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\begin{document}
\title{Virial Relations for Nonlinear Wave Equations\newline
and Nonexistence of Almost-Periodic Solutions}
\author{ Randall Pyke\thanks{e-mail: pyke@math.toronto.edu}
\\ Department
of Mathematics, University of Toronto}
\date{March, 1996}
\maketitle
\begin{abstract}
We present a formalism for constructing integral identities
involving time almost-periodic solutions of nonlinear wave equations.
As an application we prove a nonexistence theorem
that is applicable to a large class of nonlinearities.
\end{abstract}
\section{Introduction}
A virial relation is an integral identity involving
the solution of a differential equation. The identity can be
derived from the differential equation itself
by multiplying the equation by a combination of derivatives of the
solution and then integrating over the range of the independent
variables
or, if the equation can be formulated as a variational problem,
from infinitesimal
variations of the action functional associated to the equation.
A well known
example from physics
relates the
time-average kinetic and potential energies of an $n$-particle
system under the influence of central forces (usually referred to as
the virial theorem;\ see for example [LL]).
In mathematics virial relations have been used extensively, for
example, in
deriving necessary conditions for the existence of solutions of
differential equations beginning with the work of Pohozaev [Po].
In this article we present a systematic approach to deriving virial
relations for
almost periodic solutions
of nonlinear wave equations (NLW's) of the form
\begin{equation}
\partial_{t}^{2}\varphi - \Delta\varphi + f(\varphi) = 0.
\end{equation}
Here $\varphi:\;\RR_{x}^{N}\times\RR_{t}\rightarrow\RR$,
\ $\partial_{t}^{2}\varphi = \partial^{2}\varphi/\partial t^{2}$,
\ $\Delta\varphi = \sum_{i=1}^{N}\partial^{2}\varphi/
\partial x_{i}^{2}$,
and $f:\;\RR
\rightarrow\RR$ with $f(0) = 0$.
By almost periodic solution we understand solutions that
are almost periodic in time $t$ and $L^{2}(\RRn)$ in space $x$.
Almost periodic solutions have special significance in field
theories in that they
represent bound states, i.e., solutions that are localized
in space uniformly in time (cf. below).
Our objective is twofold.
First, we want to illuminate a method
that has been known and used
in various guises for many years in mathematics and physics and
is well-suited for problems arising in nonlinear differential
equations.
Second, we apply this method to derive necessary
conditions for the existence of almost periodic solutions of
NLW.
We remark about the notion of a periodic solution.
Due to the invariance of NLW under the Poincar\'{e} group,
an almost periodic solution may not appear to be almost periodic
in another Lorentz frame, although it still satisfies NLW in this
frame.
For example, if the second frame is one which is moving
at a constant (nonzero) velocity with respect to the first frame
(in which the solution is almost periodic),
then the solution is not almost periodic in this moving frame.
This means that when we are considering an almost periodic solution
we are tacitly
referring to a particular Lorentz frame
(it is not unique).
Thus, for us an almost periodic
solution of NLW is a solution that is almost periodic in {\em some}
Lorentz frame.
In this paper we are concerned solely with almost periodic solutions
and do not address the question of the
existence of general solutions.
In particular, our results are
that NLW's with certain nonlinearities do not possess
almost periodic
solutions.
Never the less,
global existence theory does apply to
some of these equations (for a survey see [GV] or [Str2]).
Thus, together these two results imply that there are NLW's that
possess global (in time) solutions but that none of these
solutions are almost
periodic \ -\ that the set of solutions of
these NLW's do not contain functions
with a certain temporal behavior
(almost periodicity).
An example of such a NLW is given below (cf. Example 1 in Section 3).
When NLW is
viewed as an evolution equation in time it is natural to
attempt to characterize
solutions by their temporal behavior. The work of this article
is a start in that direction. In deciding what sorts
of behavior are appropriate for such a characterization, we are
motivated by two observations. The first is that
\ $L^{2}(\RRn)$-valued almost periodic functions
have the property of being uniformly bounded with respect to time.
That is, for any
$\ep>0$ there exists a ball $B(\rho)\subset\RRn$ of radius
$\rho = \rho(\ep)$ such that
\begin{equation}
\int_{B^{c}(\rho)}|\varphi(x,t)|^{2}\,dx\;<\;\ep
\end{equation}
for all $t$
(cf. Definition 4.5 and Lemma 4.7 below).
We refer to functions satisfying this condition as being
{\em bound states}.
Secondly, an important theorem in quantum
mechanics due to Ruelle [R] states that bound state
solutions of the Schr\"{o}dinger equation are almost periodic in
time.
The converse is true as we have just seen.
Thus, for the Schr\"{o}dinger equation bound states are
characterized by their temporal behavior and by this alone.
We believe that a similar statement can be made
for some nonlinear equations such as NLW and the nonlinear
Schr\"{o}dinger equation [Py].
That is, solutions of these equations that are bound states, as
defined above,
are almost periodic in time (we may have to first transform to
an appropriate coordinate system via a symmetry of the equation).
Therefore, from this perspective the
work presented in this article
aims to contribute towards a classification of nonlinear
wave equations according to whether they possess bound states or
not.
One of the few NLW's known to possess
periodic solutions that are localized in $x$ (often referred to as
"breathers") is the sine-Gordon equation\ $\partial_{t}^{2}\varphi
-\partial^{2}_{x}\varphi + \sin(\varphi) = 0$.
Here the solutions are given by the formula
\begin{equation}
\varphi(x,t;\omega)\;=\;4\tan^{-1}\left(\frac{\ep\sin \omega t}
{\omega\cosh\ep x}\right),\;\;\;\;\ep^{2} + \omega^{2} = 1.
\end{equation}
Being an integrable system, these solutions can be found using
techniques related to the inverse scattering transform [MNPZ].
However, many NLW's of interest are not integrable, for example, the
"$\varphi^{4}$" equation; \ $\partial_{t}^{2}\varphi -
\partial^{2}_{x}\varphi
+ \varphi - \varphi^{3} = 0$, \ which
is an important model in particle physics.
It is not known whether this latter equation possesses periodic
solutions.
The method we describe in this article
does not depend on the integrability
properties of the equation.
\vspace{.2in}
We briefly review previous studies concerning periodic solutions of
nonlinear wave equations.
The structural
stability of the sine-Gordon breather has been an object of study
for a number of years (for recent work see [BMW] and [D], for an
earlier study
see [MS]).
Here one looks for periodic solutions of the perturbed
sine-Gordon equation
that are close to the sine-Gordon breather.
Because the sine-Gordon breather
is known explicitly, a detailed analysis
can be carried out.
These references reveal a complex interaction between the breather and
nonlinearity $\sin(\varphi)$
that is easily upset by a perturbation.
The conclusion is that typically the perturbed equation
does not have a periodic solution close to the sine-Gordon breather.
This result is
corroborated by the work [Ki] in which the sine-Gordon equation is
singled-out as essentially the only NLW on
$\RR^{1+1}$ having (analytic) breathers.
In the same vein one can look at how periodic solutions of
{\em linear} equations behave under nonlinear perturbations.
This was investigated in [Si] where it
was shown that these solutions, like
the sine-Gordon breather, are
generically unstable under nonlinear structural perturbations.
A necessary condition for the existence of periodic solutions on
$\RR^{1}$ follows from a result of Coron's [Co] which states that
if $\varphi\in C^{2}(\RR,\RR)$ is a $2\pi/\omega$ -periodic
solution of NLW, then $\omega^{2}\leq f'(0)$. Recently, we have been
able to extend this result to multi-spatial dimensions [PS].
The situation is very different on a bounded or semi-bounded
spatial domain;\ here periodic and almost periodic
solutions are abundant.
In [SV],[V2],[V3], and [We],
periodic, quasiperiodic, or almost periodic solutions of NLW
on the half-line $\RR_{+}$ were
constructed using methods from centre manifold theory.
Here the one dimensionality of the spatial variable is essential
as NLW is treated as a dynamical system in a phase space of
periodic functions with $x$ playing the role of the dynamical
variable. The (one-sided) exponentially decaying
periodic solutions are points in the
stable or unstable manifolds of the zero solution.
The failure of these manifolds to intersect prohibits extending these
existence results to the entire real line [V3].
\ [Sc] and [Sm] apply a similar analysis for radially symmetric
periodic solutions on $\RR^{N+1}$.
For NLW on an interval, [CW] and [Wa] used
infinite dimensional KAM theory
to prove the existence of families of periodic solutions,
while [BCN] applied variational methods
and [H] used bifurcation theory
to prove the existence of periodic solutions.
\vspace{.2in}
Although the above results indicate that almost periodic solutions of
nonlinear wave equations on $\RR^{N+1}$
are rare, the question of existence is
still very much open.
With this in mind we set out to investigate what
conditions the nonlinearity must satisfy in order for NLW on the
infinite spatial domain $\RRn$ to
support solutions that are time periodic or,
more generally, time almost
periodic.
Although in this paper
we will be considering the unbounded spatial domain, we could also
carry out our analysis on bounded or semi-bounded
spatial domains.
Our approach to this problem was motivated by the study [V1]
where the periodic NLW on $\RR^{1+1}$
was formulated as a variational problem.
Since periodic solutions are critical points
of the action functional,
the Fr\'{e}chet derivative of the
functional
is zero at these points. Evaluating the Fr\'{e}chet derivative on
certain functions derived from the solution itself results in
integral identities
involving the solution and nonlinearity.
>From these identities several nonexistence theorems for
classical (i.e., $C^{2}(\RR^{2})$) periodic solutions was proven.
Subsequently, the development of our ideas
in the direction of describing virial relations as
infinitesimal variations of the action functional
(cf. the appendix to the present paper) was aided by a recent
article [M] which surveys the role of virial relations in physical
theories that are based on an action principle.
The work presented in this paper
extends the results of [V1] in several
directions. First, we work with more general
quasiperiodic solutions (multiple, but finitely many, frequencies
instead of one) and in
several spatial dimensions. Our admissible set of solutions is
also larger, being of class $H^{1}$ in space and time (weak
solutions).
After deriving a class of virial relations
for quasiperiodic solutions and using these to state
necessary conditions for the existence of
such solutions, we will see
that by using ideas from ergodic theory
we can express our results in a way that makes sense for the
larger class of almost periodic solutions (i.e., allowing for
infinitely many frequencies).
In the last section of this paper
we show that these formulae are in fact
valid for almost periodic solutions.
Historically, the idea of formulating a nonlinear equation as a
minimization problem to obtain necessary conditions for the existence
of solutions has its origins in a work of Pohozaev [Po] who used
the method to study solutions of a nonlinear elliptic boundary
problem.
Subsequent applications of this technique can be found, for
example, in [Str1] and
[B] where it was used to study stationary states of some nonlinear
wave equations, and in [BL] and [BrLi] where it was used
in a study of a class
of nonlinear elliptic variational problems.
\vspace{.2in}
The main results of this article are the following.
We first derive a class of integral identities that must be
satisfied by almost periodic solutions of NLW
(Theorems 2.7 and 4.10).
Then, by choosing a particular subclass of these
we are able to prove the
following theorem, which we state here without all the
technical details (cf. Theorems 3.5 and 5.1).
\vspace{.2in}
\noindent{\bf Theorem}\ \ {\em Suppose } $\varphi$ {\em is an
almost periodic solution of NLW such that } $\varphi(\cdot,t)
\in H^{1}(\RRn)$. {\em Let } $F(z)=\int_{0}^{z}f(w)\,dw$.
{\em If } $F(z) - czf(z)\leq 0$ {\em for some } $c\in
[\frac{N-2}{2N},\frac{1}{2}]$ {\em and for all } $z$ {\em such
that } $|z|\;\leq\;\|\varphi\|_{L^{\infty}(\RR^{N+1}) }$,
{\em then } $\varphi$ {\em is independent of time}.
\vspace{.2in}
Application of this theorem can be widened by considering small
amplitude solutions since then only properties of the nonlinearity
near zero enters. For such solutions we present two nonexistence
results (Corollaries 3.6 and 3.7).
The first has relevance, in particular, to nonlinear
Klein-Gordon equations (i.e., $f'(0)>0$) with odd nonlinearity.
For the second we combine Wirtinger's inequality
with virial relations
to obtain a result similar to Coron's [Co]
in multi-spatial dimensions.
By taking
advantage of the decay of solutions as $|x|\rightarrow\infty$
we can, by "localizing" the virial relation in a neighborhood of
spatial infinity, extend some of the results about small
amplitude solutions to solutions of
arbitrary amplitude (Corollary 3.8).
\vspace{.2in}
Before concluding this introduction we present some heuristics to
help motivate and describe our approach.
We begin with the observation that a
function $\varphi(x,t)\in$\mbox{$H^{1}(\RR^{N+1})$}
can be viewed
as a
function from $\RR$ to $H^{1}(\RRn)$ via the map $t\mapsto
\varphi(\cdot,t)$.
Furthermore, if $\varphi$ is $2\pi/\omega$ -periodic then we can
write $\varphi(x,t) = \gamma(x,\omega t)$ where
$\gamma:\;S^{1}\rightarrow H^{1}(\RRn)$ is a function defined on
the unit circle $S^{1}$.
>From this it is clear how to define quasiperiodic functions:
\ $\varphi$ is quasiperiodic if $\varphi(x,t) =
\gamma(x,\omega t)$ for some $\gamma:\ \TT^{l}\rightarrow
H^{1}(\RRn)$ where $\TT^{l}$ is the $l$-torus
$S^{1}\times\cdots\times S^{1}$, and for some $\omega\in\RR^{l}$.
We call $\gamma$ the {\em generating function} of $\varphi$ and
$\omega$ the {\em frequency} of $\varphi$.
Given a quasiperiodic solution $\varphi$ of NLW with frequency
$\omega\in\RR^{l}$, by the
chain rule we
derive an equation satisfied by its generating function
$\gamma$;
$$
{\cal D}_{\omega}^{2}\gamma - \Delta\gamma + f(\gamma) = 0.
$$
Here ${\cal D}_{\omega}^{2} = \sum_{i,j = 1}^{l}
\omega_{i}\omega_{j}{\cal D}_{i}{\cal D}_{j}$,
\ ${\cal D}_{i} = \partial/\partial\theta_{i}$,
and $\theta_{1},\ldots,\theta_{l}$ are coordinates on $\TT^{l}$.
We call this equation the {\em nonlinear wave equation on } \ $\RRn
\times\TT^{l}
\equiv\Omeg$ with frequency $\omega$.
To derive virial relations for quasiperiodic solutions $\varphi$ then,
we derive virial relations for solutions $\gamma$ of NLW
on $\Omeg$ and transfer these back to $\varphi$ using
the identification $\varphi(x,t) = \gamma(x,\omega t)$.
In general, an almost periodic function is a quasiperiodic function
with infinitely many (independent)
frequencies. Almost periodic
functions $\varphi$ can be characterized
by generating functions $\gamma$
that are defined on the infinite dimensional
torus $\TT^{\infty}$;
\ $\gamma:\;\TT^{\infty}\rightarrow H^{1}(\RRn)$.
Then, there is a dense embedding
\ $\Gamma:\;\RR\rightarrow\TT^{\infty}$
such that $\varphi(t) = \gamma(\Gamma(t))$.
For almost periodic solutions though,
we will work directly with
the function itself rather than with its generating function. This is
facilitated by passing from $\TT^{\infty}$ to $\RR$ via the
Bohr compactification of the real line which is realized by the
formula
\begin{equation}
\int_{\TT^{\infty}}\gamma(\theta)\,d\theta\;=
\;\ergint \Big(\gamma\circ\Gamma\Big)(t)\,dt.
\end{equation}
Here $d\theta$ is the (normalized) Haar measure on $\TT^{\infty}$.
That is, instead of integrating $\gamma$ over the torus
$\TT^{\infty}$, which is its
space average, we take its time average along the curve $\Gamma$.
To illustrate the general strategy as well as to describe some
aspects of our approach, we present an example of a virial relation.
Suppose $\gamma$ is a solution of NLW on $\Omeg$ with
frequency $\omega$ and let $\beta:\;\TTl\rightarrow H^{1}(\RRn)$ be
some function on $\TTl$.
Then, multiplying NLW by $\beta$
and integrating we have,
\begin{displaymath}
\int_{\TTl}\int_{\RRn} \beta\Big({\cal D}^{2}_{\omega}\gamma
- \Delta\gamma
+ f(\gamma)\Big)\;\;=\;\;0.
\end{displaymath}
Integrating by parts we obtain
$$
\int_{\TTl}\int_{\RRn}\Big\{
-\Dw\gamma\Dw\beta + \n\gamma\cdot\n\beta
+ f(\gamma)\beta\Big\}\;=\;0,
$$
which expresses that $\gamma$ is a weak solution.
Now take for the multiplier $\beta$ some
combination of $\gamma$ and its derivatives, for example,
\ \ $\beta = x\cdot\n\gamma$.
Using this $\beta$ in the latter formula and removing a divergence
term we arrive at the virial relation
\begin{equation}
\int_{\TTl}\int_{\RRn}
\Big\{\frac{1}{2}(\Dw\gamma)^{2} +
\Big(\frac{2 -N}{2N}\Big)\mid\n\gamma\mid^{2}
- F(\gamma)\Big\}\;\;=\;\;0,\;\;\;\;F' = f.
\end{equation}
We see right away that
in spatial dimensions $1$ or $2$, \ $F$ cannot be a
nonpositive function. Thus, we have a necessary condition
on the nonlinearity for the existence of solutions of NLW on
$\Omeg$.
The virial relation for the quasiperiodic solution $\varphi$
can be recovered from this by the identification
$\varphi(x,t) = \gamma(x,\omega t)$ and where the integral over
$\TTl$ is replaced by the time-mean over $\RR$ (cf. (1.4)).
What is required to make this argument rigorous is that
$\gamma$ be sufficiently regular and integrable.
How regular and integrable depends on how $\beta$ is defined.
One could, at the start, consider only those solutions for which
the formal manipulations remain
valid.
However, the final formula (the virial relation) that one
obtains requires less of the solution
than the rigorous analysis asks for.
In the above example we see that the virial relation makes sense
(i.e., the integral in (1.5) is finite)
if $\gamma\in H^{1}(\Omeg)$
and $F(\gamma)\in L^{1}(\Omeg)$.
In our analysis we assume from a solution and nonlinearity
only
what is needed to make the (formally derived) virial
relation a well-defined
formula. This defines then the largest possible class of
solutions (for a given nonlinearity) for which the virial relation
is valid. To make the derivation rigorous we
regularize the multiplier $\beta\mapsto\beta_{\ep}$,
\ $\beta_{0} = \beta$, derive the virial relation corresponding
to $\beta_{\ep}$,
and then regain the virial relation
corresponding to $\beta$ in the limit $\ep\rightarrow 0$.
Because we allow for infinitely many frequencies in the almost
periodic case,
the construction of virial relations for almost periodic
solutions differs from that
for periodic and quasiperiodic solutions.
But since
periodic and quasiperiodic solutions are also almost periodic, the
latter results include the former. However, the
method of proof for
quasiperiodic solutions is different than the proof for almost
periodic solutions.
We have included both proofs here because the
method for
quasiperiodic solutions could be adapted to handle
other partial differential equations; those defined on open subsets
$\Omega$ of $\RR^{m}$ (bounded or unbounded) where the
solutions are in $L^{2}(\Omega)$.
The technical tools for this are a regularization of the solution
and Lebesgue's dominated convergence theorem. The dominated
convergence theorem cannot be
applied to almost periodic solutions because they are not
square integrable on $\RR^{N+1}$.
Instead, we utilize the uniform boundedness of almost periodic
functions (cf. (1.2)) in conjunction
with the uniform (in $t$) convergence
of $\beta_{\ep}$ on compact subsets of $\RRn$.
In the above example we used the function $\beta = x\cdot\n\gamma$
to derive
a virial relation. One can view $x$ as
a vector field $v$ on $\RRn$:\ $v(x) = x$.
Conversely, for any vector field $v$ we may derive a
virial relation by setting $\beta = v\cdot\n\gamma$. In our
study we use this identification to characterize a class of virial
relations.
Furthermore, these vector fields generate flows
$\Phi_{\lambda}:\;\RRn\rightarrow\RRn$ which in turn define
transformation groups $T_{\lambda}$ acting on functions via the
formula $T_{\lambda}\varphi\equiv\varphi\circ\Phi_{\lambda}$.
In general, every transformation
group will lead to a virial relation via the infinitesimal
variations of the action associated to NLW under this group.
If $T_{\lambda}$ is a symmetry of the action then there is a
corresponding differential identity (a conservation law:\ Noether's
theorem) which can be thought of as a "trivial" virial relation.
This is discussed more thoroughly in the appendix.
\vspace{.2in}
We outline the contents of this paper. The next section deals with
quasiperiodic solutions of NLW.
We define quasiperiodic functions and the types of
quasiperiodic solutions we will be concerned with, as well as
discuss
our hypothesis on the nonlinearity.
Then we state and prove the main result of that section (Theorem 2.7):
\ a class of virial relations for quasiperiodic solutions of NLW
that are characterized by vector fields on $\RRn$.
In the following section, Section 3, we use these virial relations
to derive
several nonexistence
results.
Sections 4 and 5 deal with more general almost periodic solutions.
The main results there, Theorems 4.10 and 5.1,
characterize a class of virial relations for almost periodic
solutions by vector fields on $\RRn$ and state a necessary
condition for the existence of such solutions.
In the appendix we show how virial relations
may be derived by formulating NLW as a variational problem.
This will elucidate the relationship
between virial
relations and conservation laws.
\vspace{.2in}
\noindent \underline{Notation:}\ \
$\n$ denotes the gradient operator on
$\RRn$:\ \ $\n = (\partial_{1},...,
\partial_{N}),\;\;\partial_{i} = \partial/\partial x_{i},$
and for multi-index $a\in \ZZ^{N}\geq 0,\;\;\partial^{a} = \partial
_{1}^{a_{1}},...\partial_{N}^{a_{N}}$.
By the vector $\omega\in\RR^{l}$ we
will always mean an $l$-tuple of incommensurate
numbers ("frequencies").
${\bf 1}$ will stand for the identity matrix.
$\TTl$ will denote the $l$-torus $S^{1}\times\cdots\times S^{l}$,
and $\Omeg \equiv \RRn\times\TTl$.
Given a frequency vector $\omega\in\RR^{l}$ we define the differential
operator $\Dw\equiv\omega\cdot{\cal D}$ on $\TT^{l}$
where $\D = (\D_{1},\ldots,\D_{l})$,\
\ $\D_{i} = \partial/\partial\theta_{i}$, is the gradient operator
on $\TTl$.
\ $\|\psi\|_{p}$ is the $L^{p}(\RR^{n})$ norm and $H^{s,p}
$ the $L^{p}$ Sobolev space of order $s$ which for $p=2$ we write
simply as $H^{s}$. The space $L^{p}\Big(\RR,\,L^{q}(\RRn)\Big)$
denotes the set of functions $\varphi:\;\RR\rightarrow L^{q}
(\RRn)$ such that $\int_{\RR}\|\varphi(t)\|^{p}_{L^{q}(\RRn)}
\equiv \|\varphi\|_{p,q}^{p} < \infty$.
$C^{\infty}_{c}(\RRn)$ denotes smooth functions on $\RRn$
with compact
support. $(L^{q}(\RRn))^{n}$ is the set of $n$-dimensional
vector valued functions
on $\RRn$, each component being an element of $L^{q}(\RRn)$.
For $v:\RRn
\rightarrow\RRn$, $dv = dv(x)$
denotes the matrix-valued function with entries $[dv]_{ij}=
\partial_{i}v^{j}$; then,\ \ $\nabla\cdot v = tr\,dv$
where $tr\,A$ is
the trace of the matrix $A$.
\vspace{.2in}
\noindent\underline{Acknowledgements:}\ \ I would like to
thank Professors I.M. Sigal and I. Kupka for stimulating and
helpful discussions. Financial support from NSERC of Canada, the
Ontario Ministry of Education, and the University of Toronto is
gratefully acknowledged.
\section{Virial relations for quasiperiodic solutions}
\setcounter{equation}{0}
\subsection{Quasiperiodic solutions}
\begin{defn} Let $\Omeg\equiv\RRn\times\TTl$. \ $\varphi:\;\RR^
{N+1}\rightarrow \RR$
is an $l$-{\em quasiperiodic function with frequency
$\omega\in\RR^{l}$}
if $\varphi(x,t) = \gamma(x,\omega t) = \Big(\gamma\circ\Gamma_{\omega}
\Big)
(t)$
for some function $\gamma\in C(\Omeg)\cap H^{1}(\Omeg)$ where
$\Gamma_{\omega}:\RR\rightarrow\TT^{l}$ is the continuous dense
embedding of $\RR$ into $\TT^{l}$ given by \ $t\mapsto (\omega t)
\;mod\,2\pi$.
We call $\gamma$ the
ENDBODY
{\em generating function} of $\varphi$.
\end{defn}
To fix what we mean by a classical
quasiperiodic solution of NLW we state the following definition.
\begin{defn}
An $l$-quasiperiodic function
$\varphi:\;\RR^{N+1}\rightarrow\RR$ is a {\em classical quasiperiodic
solution of NLW on $\RR^{N+1}$} (equation (1.1)) if
$\varphi$ solves NLW and
its generating function is of
class $C^{2}(\Omeg)\cap H^{2}(\Omeg)$.
\end{defn}
\noindent Why we have included the integrability property $H^{2}(
\Omeg)$ of $\gamma$ will be made clear below.
To motivate our notion of a weak quasiperiodic solution of NLW
we first
derive an equation satisfied by the generating
function of a classical quasiperiodic solution.
\setcounter{lemma}{2}
\begin{lemma}
Suppose $\varphi$ is a classical $l$-quasiperiodic solution
of NLW
with frequency $\omega$.
Then its generating function $\gamma$ solves the equation
\begin{equation}
\Dw^{2}\gamma -
\Delta\gamma + f(\gamma)
\;=\;0
\end{equation}
on $\Omeg$.
Here ${\cal D}^{2}_{\omega}\gamma = \sum_{i,j=1}^{l}\omega_{i}
\omega_{j}{\cal D}_{i,j}^{2}\gamma$.
Conversely, if $\gamma$ is a member of $C^{2}(\Omeg)
\cap H^{2}(\Omeg)$ and
solves (2.1), then the function
$\varphi(x,t)=\gamma(x,\omega t)$ is a classical l-quasiperiodic
solution of NLW with frequency $\omega$.
\end{lemma}
\noindent Proof:
The curve $\Gamma = \{(x,(\omega t)
\,mod\, 2\pi)\;\mid\; (x,t)\in\RR^{N+1}\}$ is dense in
$\Omeg$ and by the chain rule $\gamma$ solves (2.1) along
$\Gamma$. By the continuity of
$\gamma,\Delta\gamma,\D^{2}_{ij}\gamma$
and $f$ we conclude that $\gamma$ solves (2.1) on
$\Omeg$.
The converse is just the chain rule.\ \ $\Box$
\vspace{.2in}
\setcounter{defn}{3}
\begin{defn}
Equation (2.1) is the {\em nonlinear wave equation (NLW) on} $\Omeg$
\mbox{{\em with frequency} $\omega$.}
\end{defn}
Equation (2.1) motivates our definition of a weak solution of NLW
on $\Omeg$.
\begin{defn} \ $\gamma:\Omeg\rightarrow\RR$
is a {\em weak solution of NLW on} $\Omeg$ {\em with frequency}
$\omega$
if $\gamma\in H^{1}(\Omeg)$
and
\begin{equation}
\int_{\TT^{l}}
\int_{\RR^{n}}\Big\{
-\Dw\gamma\Dw\beta
\;+\;
\nabla\gamma\cdot\nabla\beta\;+\;
f(\gamma)\beta\Big\}\;\;\;\;=\;\;\;\;0\;\;\;\;\;\;\forall\beta\in
H^{1}(\Omeg).
\end{equation}
\end{defn}
\begin{defn}
A quasiperiodic function
$\varphi:\RR^{N+1}\rightarrow\RR$ is a {\em weak $l$-quasiperiodic
solution of NLW on $\RR^{N+1}$ with frequency} $\omega$
if its generating function $\gamma$ is a weak
solution of NLW on $\Omeg$ with frequency $\omega$.
\end{defn}
Using standard arguments
from the calculus of variations, if $\gamma\in C^{2}(\Omeg)\cap
H^{2}(\Omeg)$ then (2.1)$\Leftrightarrow$(2.2).
This is where the integrability prÿoperty $H^{2}(\Omeg)$
enters:\ \ it assures us that the boundary terms that arise when
integrating by parts when going from (2.1)$\Rightarrow$(2.2) vanish.
\vspace{.2in}
We will work with the generating function
$\gamma$ from now on in this section, deriving
virial relations for solutions of NLW on $\Omeg$.
The virial relations for solutions $\varphi$ of NLW on
$\RR^{N+1}$ can be recovered
from these through the identification $\varphi(x,t) =
\gamma(x,\omega t)$.
\vspace{.2in}
\noindent{\bf Conditions on the nonlinearity}
\vspace{.1in}
For the derivation of virial relations for solutions $\gamma$
of NLW on $\Omeg$ we require
that $f$ be continuous with $f(0) = 0$, and that $f(\gamma)
\in L^{2}(\Omeg)$ and $F(\gamma)\in L^{1}(\Omeg)$ where
$F(z)\equiv\int_{0}^{z}f(w)\,dw$.
The latter two conditions
will depend on two factors, the growth of $f$ and the
integrability of $\gamma$.
If $\mid\! f(z)\!\mid \;\leq \;c(
\mid\! z\!\mid + \mid\! z\!\mid^{q/2})$
for some $q\in [2,\infty)$,
then $f:\;L^{2}(\Omeg)\cap L^{q}(\Omeg)\rightarrow L^{2}(\Omeg)$
and $F:\;L^{2}(\Omeg)\cap L^{q}(\Omeg)\rightarrow L^{1}(\Omeg)$.
This is true by virtue of the inequalities
\begin{eqnarray*}
\mid f(z)\mid^{2} &\leq& c_{1}\mid z\mid^{2} +
c_{2}\mid z\mid^{(q+2)/2} + c_{3}\mid z\mid^{q}, \\
\mid F(z)\mid &\leq& c_{4}\mid z\mid^{2} + c_{5}\mid z
\mid^{(q+2)/2}
\end{eqnarray*}
and the fact that $L^{2}(\Omeg)\cap L^{q}(\Omeg)\subset L^{q'}(\Omeg)
$ for all $q'\in [2,\;q]$.
If $f$ satisfies a
Lipschitz condition at the origin, then $f: L^{2}(\Omeg)
\cap L^{\infty}(\Omeg)\rightarrow L^{2}(\Omeg)$ and
$F: L^{2}(\Omeg)\cap L^{\infty}(\Omeg)\rightarrow L^{1}(\Omeg)$.
Indeed, let $c$ and $d$ be such
that $\mid\! f(z)\!\mid< c\mid\! z\!\mid$ for $\mid\! z\!\mid2$.
Therefore, because we are assuming that $\gamma\in H^{1}(\Omeg)$,
a priori the parameter $q$ in (H1) can be any number in the interval
$[2,\infty)$ in the case $N+l=2$, or any number in the interval
$[2,\,2(N+l)/(N+l-2)]$ in the case $N+l>2$.
We include the parameter $q$ in the hypothesis (H1) so as to
make special consideration
when the solution $\gamma$
possesses additional integrability than that given by the
Sobolev embedding.
\subsection{Virial relations for quasiperiodic solutions}
A virial relation involving the solution $\gamma$
of NLW on $\Omeg$ results when in
equation (2.2) we take for $\beta$ some combination of $\gamma$
and its derivatives.
A class of such virial relations can be characterized by vector
fields
on $\RRn$ (the spatial domain): given a vector field $v$,
set $\beta = v\cdot\nabla\gamma$. The resulting virial
relation is the content of the following theorem.
\setcounter{thm}{6}
\begin{thm}
Let $N$ and $l$ be positive integers, $\omega\in\RR^{l}$
any vector of incommensurate frequencies,
and $v$ a smooth
vector field on $\RRn$ that satisfies
\begin{equation}
\sup_{x\in\RRn}\mid
\partial^{a}v(x)\mid\mid x\mid^{\mid a\mid -1} < \infty
\;\;\;\;\;\;\;\mid a\mid = 0,1,2.
\end{equation}
If
(i)\ \ $N = l = 1$ with $\gamma\in H^{1}(\Omega_{1,1})$ satisfying
(2.2)
and
$f$ satisfying (H1) for some (any) $q\in[2,\infty)$, or
(ii)\ \ if $N+l>2$ and if $\gamma\in H^{1}(\Omeg)\cap L^{q}(\Omeg)$,
for some $q\in [2,\infty]$, satisfies (2.2)
and $f$ satisfies (H1) with this
same $q$,
\noindent
then
\vspace{.2in}
\begin{equation}
\int_{\TT^{l}}
\int_{\RRn}
\left\{ tr\,dv\left(\frac{1}{2}
(\Dw\gamma)^{2}
- F(\gamma)\right)\;+\;
\nabla\gamma\cdot\big[dv-\frac{1}{2}tr\,dv\,{\bf 1}\big]
\nabla\gamma\right\}
\;\;=\;0,
\end{equation}
\vspace{.2in}
\noindent where $F(z)\equiv\int_{0}^{z} f(w)\,dw$.
\end{thm}
\vspace{.2in}
\noindent\underline{Remarks:}\ \ Equation (2.4) is the virial relation
for the solution $\gamma$ of NLW on $\Omeg$ associated to the
vector field $v$. The corresponding virial relation for $\varphi$, where
$\varphi(x,t) = \gamma(x,\omega t)$, is
\begin{displaymath}
\ergint\int_{\RRn}
\left\{tr\,dv\; \big(\frac{1}{2}(\partial_{t}\varphi)^{2}
- F(\varphi)\big)\;+\;
\;\nabla\varphi\cdot\big[dv-\frac{1}{2}tr\,dv{\bf 1}\big]\,\nabla\varphi
\right\}\;\;=\;\;0.
\end{displaymath}
The derivation of this formula from equation (2.4)
follows from some results of ergodic theory and will be presented in
Section 4.
\vspace{.2in}
During the proof of Theorem 2.7 we will require the following Lemma.
\setcounter{lemma}{7}
\begin{lemma}
\ \ If
$h\in(H^{1,1}(\RRn))^{n}$ then $\int_{\RRn}\n\cdot h\;=\;0$.
\end{lemma}
\noindent\underline{Proof of Lemma 2.8}:
There exists a family $\{h_{\ep}\}\subset(C^{\infty}_{c}(\RRn))^{n}$
that converges to $h$ in
$(H^{1,1}(\RRn))^{n}$ as $\ep\rightarrow 0$.
In particular this means that
$\lim_{\ep\rightarrow 0}\int_{\RRn}\n\cdot h_{\ep}\;=\;\int_{\RRn}
\n\cdot h$.
For each $h_{\ep}$ we have, via the divergence theorem,
that $\int_{\RRn}\n\cdot h_{\ep} = 0$. Therefore
$\int_{\RRn}\n\cdot h = 0$\ \ \ \ $\Box$
\vspace{.3in}
\noindent
\underline{Proof of Theorem 2.7}:
Our motivation in deriving equation (2.4)
comes from proceeding formally:\ \ take
$\beta = v\cdot\n\gamma$
in equation (2.2) and apply the standard rules of calculus (without
justification) to arrive at equation (2.4).
In the proof below we show that
this procedure can be made rigorous after a
suitable regularization of $\gamma$ and $v$.
To this end we introduce the family of operators
$\Rrep = \Rrep(-\ep\,i\n)$ on $L^{2}(\RRn)$ defined through
the Fourier
transform:\ \ $\widehat{\Rrep\psi}
= (1+\ep^{2}p^{2})^{-1}\hat{\psi}(p)$.
Because $R_{\ep}$ dampens the high frequency
components of $\hat{\psi}$,
it acts as a smoothing operator.
We have that
$R_{\ep}\psi\rightarrow\psi$ in $L^{2}(\RRn)$
as $\ep\rightarrow 0$ with $\|R_{\ep}\psi\|_{2}
\leq\|\psi\|_{2}$, and that $\Rrep$ commutes with $\n$.
Using the Fourier transform we see that $\Rrep$ acts as a bounded
self-adjoint operator from $H^{s}(\RRn)$ to $H^{s+2}(\RRn)$.
We begin by regularizing $v\cdot\n\gamma$.
For $\ep,\delta>0$
define
\begin{equation}
\beta_{\ep,\delta}\equiv
v_{\delta}\cdot\nabla_{\ep}\gamma
\end{equation}
where
\begin{equation}
v_{\delta}(x)=\frac{v(x)}{1+\delta\mid x\mid^{2}}
\end{equation}
(this controls the growth of $v$ at infinity) and
\begin{equation}
\nabla_{\ep}\equiv \nabla\R2rep
\end{equation}
(which smooths $\gamma$).
By condition (2.3), $v_{\delta}\in L^{\infty}(\RR^{n})$
for $\delta>0$ while for $\mid\!a\!\mid> 0$,
\ \ $\partial^{a}\vdi\in L^{\infty}(\RRn)$ with
$\|\partial^{a}\vdi\|_{\infty}$ bounded uniformly in $\delta$.
The end result of the regularization (2.5) is that $\n\gamma$ gains
an additional derivative while retaining its integrability
properties even under multiplication by $\vd$. This is
precisely what is required to make the formal manipulations
rigorous.
\vspace{.2in}
Since $\beta_{\ep,\delta}\in H^{1}(\Omeg)$ for $\ep,\delta >0$,
we have
(cf. equation (2.2))
\begin{equation}
\int_{\TT^{l}}
\int_{\RR^{n}}
\Big\{-\Dw\gamma\Dw\beta_{\ep,\delta}
\;+\;
\n\gamma\cdot\nabla\beta_{\ep,\delta}\;+\;
f(\gamma)\beta_{\ep,\delta}
\Big\}\;\;\;\;=\;\;\;\;0,
\;\;\;\;\;\;\;
\forall\ep,\delta >0.
\end{equation}
Formally, we derive equation (2.4) from equation (2.2)
(with $\beta = v\cdot\n\gamma$) by writing the
integrand in equation (2.2) as $g + div\,(h)$ for some
$\RR^{N+l}$ vector valued function
$h$ on $\Omeg$.
Here $div = (\n_{x},\n_{\theta})$ is the divergence operator on
$\Omeg$.
The divergence term vanishes after performing the integration
and we are left with \ $\int_{\Omeg} g = 0$;
\ this is equation (2.4).
But since we are dealing with weak solutions we begin with the
regularized equation (2.8) instead of (2.2).
To derive equation (2.4) from this, then, we will write the integrand
in (2.8)
as $\tilde{g} = g + g_{\ep,\delta} + div\,(h)
+ div\,(h_{\ep,\delta})$
where $g$ and $h$ are as before.
We will show that $\lim_{\ep,\delta\rightarrow 0}\int_{\Omeg}
g_{\ep,\delta} = 0$. Then,
since $\int_{\Omeg}div\,(h_{\ep,\delta}) = 0$
for all $\ep,\delta > 0$, we have that
$\lim_{\ep,\delta\rightarrow 0}\int_{\Omeg}\tilde{g} =
\int_{\Omeg} g$. To prove that $\lim_{\ep,\delta\rightarrow 0}
\int_{\Omeg} g_{\ep,\delta} = 0$ we first show that
when this term is integrated over $\RRn$
the resulting function of $\theta\in \TTl$ converges pointwise to
zero and can be dominated by an $L^{1}(\TTl)$ function.
Then we apply the dominated convergence theorem.
\vspace{.2in}
Consider the term (we suppress the dot product initially)
\begin{eqnarray}
\n\gamma\,\nabla\beta_{\ep,\delta} & = &
\n\gamma\,\n\, \vd\cdot\n\R2rep\gamma
\nonumber \\
& = &
\n\gamma\,\vd\cdot\n{\bf 1}\,\n\R2rep\gamma\;\;+\;\;
\n\gamma\,dv_{\delta}\,\n\R2rep\gamma.
\end{eqnarray}
In this formula, and
for similar formulae in the sequel, the expression $\vd\cdot\n
{\bf 1}$,
to the immediate left of the vector $\n\R2rep\gamma$, denotes the
$n\times n$ diagonal matrix
with diagonal entries $\vd\cdot\n$, while for
adjacent vectors the dot product is implied.
We will show that
\begin{equation}
\stackrel{lim}{\ep,\delta\rightarrow 0}
\;\;\;\int_{\TTl}\int_{\RRn}
\n\gamma\cdot\n\beta_{\ep,\delta}\;\;=
\;\;\int_{\TTl}\int_{\RRn}
\n\gamma\cdot[dv - \frac{1}{2}tr\,dv{\bf 1}]\n\gamma.
\end{equation}
Commuting $\Rrep$ through the operator $\n$
and commuting $v_{\delta}$ with $R_{\ep}$,
we write the first term on the right side of equation (2.9) as
\begin{equation}
\n\gamma\,\vd\cdot\Rrep\n{\bf 1}\,\n\Rrep\gamma\;\; =
\;\;\n\gamma\,\Rrep\vd\cdot\n{\bf 1}\,\n\Rrep\gamma \;\;-
\;\;\n\gamma\,[\Rrep,\vd]\cdot\n{\bf 1}\,\Rrep\n\gamma.
\end{equation}
Using the self-adjointness of $R_{\ep}$,
\begin{equation}
\int_{\RR^{n}}
\n\gamma\,\Rrep\vd\cdot\n{\bf 1}\,\n\Rrep\gamma \;\;=
\;\;\int_{\RRn}
(\n\Rrep\gamma)\,\vd\cdot\n{\bf 1}\,\n\Rrep\gamma\
\end{equation}
(where we recognize the integral on the left
as the inner product $\langle \n\gamma,\;
\Rrep\vd\cdot\n{\bf 1}\,\n\Rrep\gamma\rangle$ in $(L^{2}(\RRn))^{N}$).
By the chain rule,
\begin{eqnarray}
\int_{\RRn}
(\n\Rrep\gamma)\,\vd\cdot\n{\bf 1}\,\n\Rrep\gamma & =&
\int_{\RRn}\left\{\frac{1}{2}\n\cdot(\vd\mid\n\Rrep\gamma\mid^{2}) -
\frac{1}{2}\n\cdot\vd\mid\n\Rrep\gamma\mid^{2}\right\} \nonumber \\
& = & -\frac{1}{2}\int_{\RRn}
\n\cdot\vd\mid\n\Rrep\gamma\mid^{2}.
\end{eqnarray}
Here we've used Lemma 2.8 to
conclude that
$\int_{\RRn}\n\cdot (\vd\mid\n\Rrep\gamma\mid^{2}) = 0$.
Furthermore,
\begin{eqnarray}
\n\cdot\vd\,\mid\n\Rrep\gamma\mid^{2} &=&
\n\cdot v\,\mid\n\gamma\mid^{2}\;\; +
\;\;[\n\cdot\vd - \n\cdot v]\mid\n\gamma\mid^{2} \nonumber \\
& &\;\;+ 2\,\n\cdot\vd[(\Rrep - 1)\n\gamma]\n\gamma
\;\;+\;\; \n\cdot\vd[(\Rrep - 1)\n\gamma]^{2}.
\end{eqnarray}
Note that for almost all $\theta\in\TTl$,\ \ $\n\gamma
= \n\gamma(x,\theta)\in (L^{2}(\RRn))^{n}$.
Since $\n\cdot\vd\rightarrow\n\cdot v$ pointwise as
$\delta\rightarrow 0$ and $\|\n\cdot\vd - \n\cdot v\|_{\infty}$
can be bounded uniformly in $\delta$,
\ $J_{\delta}(x,\theta)\equiv
[\n\cdot\vd - \n \cdot v ]\mid\!\n\gamma\!\mid^{2}\rightarrow 0$
in $L^{1}(\RRn)$ as $\delta\rightarrow 0$ for
those $\theta$ in which $\n\gamma\in (L^{2}(\RRn))^{n}$.
Thus the function $\bar{J}_{\delta}(\theta)\equiv\int_{\RRn}
J_{\delta}(x,\theta)$
converges pointwise to zero almost everywhere
in $\TTl$ as $\delta\rightarrow 0$ and is
dominated pointwise by $\bar{J}(\theta) =
c\|\n\gamma\|
_{2}^{2}\in L^{1}(\TTl)$.
Here $c$ is the uniform bound on $\|\n\cdot\vd - \n\cdot v\|_{\infty}$.
Lebesgue's dominated convergence
theorem then implies that $\int_{\TTl}\bar{J}_{\delta}\rightarrow 0$
as $\delta\rightarrow 0$. That is, $J_{\delta}
\rightarrow 0$ in $L^{1}(\Omeg)$.
To treat the third term on the right in equation (2.14) we note that
$\|(\Rrep - 1)\n\gamma\|_{2}\rightarrow 0$ for almost all
$\theta$ while
$\|\n\cdot\vd\|_{\infty}$ is uniformly bounded in
$\delta$. Using the Schwarz inequality on $(L^{2}(\RRn))^{N}$
we see that this term, when integrated over $\RRn$,
converges to zero
in $L^{1}(\RRn)$
pointwise $\theta$-almost everywhere
as $\ep\rightarrow 0$ uniformly in $\delta$.
Since $\|\n\cdot\vd\|_{\infty}\leq\|\n\cdot v\|_{\infty} + c$, we
can dominate this term by $4(\|\n\cdot\v\|_{\infty} + c)\|\n\gamma
\|_{2}^{2}\in L^{1}(\TTl)$ and hence conclude that it
converges to zero in
$L^{1}(\Omeg)$ as $\ep\rightarrow 0$, uniformly in $\delta$.
A similar argument with the same conclusion applies to the
last term on the right hand side of equation (2.14).
\vspace{.2in}
We expand the commutator on the right hand side of equation (2.11);
\begin{eqnarray}
\int_{\RR^{n}}
\n\gamma[\Rrep,\vd]\cdot\n{\bf 1}\,\Rrep\n\gamma & = &
\int_{\RRn}
\n\gamma\Rrep\ep^{2}[\Delta,\vd]\Rrep\cdot\n{\bf 1}\,\Rrep\n\gamma
\nonumber \\
& = &
\int_{\RRn}
\n\gamma\Rrep\ep^{2}(\Delta\vd\cdot\Rrep\n{\bf 1} +
2\n\vd\n\cdot\Rrep\n{\bf 1})\Rrep\n\gamma.
\end{eqnarray}
Here we have written $[\Rrep,\vd]$ as \ $-\Rrep[(1-\ep^{2}\Delta),\vd]
\Rrep$.
The expressions $\Delta\vd$ and $\n\vd\n$ denote the
vectors $(\Delta\vd^{1},...,\Delta\vd^{N})$ and
$(\n\vd^{1}\cdot\n,...,\n
\vd^{N}\cdot\n)$ respectively.
Before estimating the two terms in this equation we first note that
for $\mid\!a\!\mid = 1,2$ the function $\mid\!p^{a}\!\mid^{2}
(1+\ep^{2}p^{2})^{-2}$ is bounded by $c\ep^{-2\mid a\mid}$,\ $c
= c(a)$,\ so that, using the Fourier transform, for $\psi\in
L^{2}(\RRn)$
\begin{equation}
\|\partial^{a}\Rrep\psi\|_{2}\leq
c\ep^{-\mid a\mid}\|\psi\|_{2},
\end{equation}
and
\begin{equation}
\lim_{\ep\rightarrow 0}
\|\ep^{\mid a\mid}\partial^{a}\Rrep\psi\|_{2}^{2}\;\;\; =
\;\;\;\lim_{\ep\rightarrow 0}
\int_{\RRn}\frac{\ep^{2\mid a\mid}\mid p^{a}\mid^{2}}
{(1 + \ep^{2}p^{2})^{2}}
\mid\hat{\psi}(p)\mid^{2}\;\;=\;\;0
\end{equation}
by dominated convergence. Using equation (2.16)
we estimate
\begin{eqnarray}
\lefteqn{\Big|\int_{\RRn}
\ep^{2}\n\gamma\Rrep\Delta\vd\cdot\Rrep\n{\bf 1}\,\Rrep\n\gamma\Big|}
\nonumber \\
& & \leq
\ep^{2}\|\n\gamma\|_{2}\;\|\Delta\vd\cdot\Rrep\n{\bf 1}\,\Rrep\n\gamma
\|_{2} \nonumber \\
& & \leq
c\ep^{2}\|\n\gamma\|_{2}\;\|\n{\bf 1}\,\Rrep\n\gamma\|_{2}
\nonumber \\
& & \leq
c\ep\|\n\gamma\|_{2}^{2}.
\end{eqnarray}
Since $\|\n\gamma\|_{2}^{2}\in L^{1}(\TTl)$ we see immediately
that the first term on the right in the last line of equation (2.15)
converges to zero in $L^{1}(\TTl)$ as $\ep\rightarrow
0$.
The constant $c$ here
is derived from $\|\partial^{a}\vd\|_{\infty}$ and is
independent of $\delta$ because these norms are uniform
in $\delta$.
For the second term on the right of equation (2.15) we have,
\begin{eqnarray}
\lefteqn{ \Big|\int_{\RRn}
\ep^{2}\n\gamma\Rrep\n\vd\n\cdot\Rrep\n{\bf 1}\,\Rrep\n\gamma\Big|}
\nonumber \\
& & \leq
\|\n\gamma\|_{2}\;\|\ep^{2}\n\vd\n\cdot\Rrep\n{\bf 1}\,\Rrep\n\gamma\|
_{2} \nonumber \\
& & \leq
c\|\n\gamma\|_{2}\;\|\ep^{2}\n^{2}{\bf 1}\,\Rrep\n\gamma\|_{2}
\;\;\rightarrow 0
\end{eqnarray}
for almost all $\theta$ as $\ep\rightarrow 0$
by (2.17), where $c$ is independent of $\delta$ (for the same
reason as before) and
$\n^{2}{\bf 1}$ denotes the matrix
$diag (\sum_{i,j=1}^{N}\partial^{2}_{ij}$).
Since $\|\ep^{2}\n^{2}\Rrep\n\gamma\|_{2}\leq c\|\n\gamma\|_{2}$
for all $\ep >0$ (cf. equation (2.16))
we can dominate the left hand side of (2.19)
by $c\|\n\gamma\|_{2}^{2}\in L^{1}(\TTl)$
to conclude that it converges to zero in
$L^{1}(\TTl)$ as $\ep\rightarrow 0$ uniformly in $\delta$.
Combining this with the preceding result (cf. equation (2.14)),
we have shown that
\begin{equation}
\lim_{\ep,\delta\rightarrow 0}
\;\;\;\int_{\TTl}\int_{\RRn}
\n\gamma\,\vd\cdot\n{\bf 1}\,\n\R2rep\gamma\;\;=\;\;
\int_{\TTl}\int_{\RRn}
-\frac{1}{2}tr\,dv\,\mid\n\gamma\mid^{2}.
\end{equation}
\vspace{.2in}
The term $\n\gamma\,dv_{\delta}\,\n\R2rep\gamma$
in equation (2.9) we write as
\begin{eqnarray}
\n\gamma\, dv_{\delta}\,\n\R2rep\gamma \;\;=\;\; \n
\gamma\,dv\,\n\gamma &+&
\n\gamma(dv_{\delta}-dv)\n\gamma \nonumber \\
&+&
\n(\R2rep -1)\gamma\,dv_{\delta}\,\n\gamma.
\end{eqnarray}
We have :\ \ $dv_{\delta}\rightarrow dv$ pointwise with
$\|dv_{\delta}- dv\|_{\infty}$ and $\|dv_{\delta}\|_{\infty}$
(matrix norm)
bounded uniformly in $\delta$, while
$\n(\R2rep -1)\gamma\rightarrow 0$ in
$(L^{2}(\RRn))^{N}$ for almost all $\theta$.
Therefore, by arguments similar to those made above,
the last two terms on the right hand
side of equation (2.21) converge to zero in $L^{1}(\Omeg)$ as
$\ep\rightarrow 0$ uniformly in $\delta$. We conclude then that
\begin{equation}
\lim_{\ep,\delta\rightarrow 0}
\;\;\int_{\TTl}\int_{\RRn}
\n\gamma\,dv_{\delta}\,\n\R2rep\gamma\;\;=\;\;
\int_{\TTl}\int_{\RRn}
\n\gamma\,dv\,\n\gamma
\end{equation}
and equation (2.10) is shown.
\vspace{.2in}
Considering now the first term in equation (2.8),
commuting $\Dw$ with $R_{\ep}$ and $\nabla$ we obtain
\begin{eqnarray}
\Dw\gamma\Dw\beta_{\ep,\delta}
& = &
\Dw\gamma\Dw\,\vd\cdot\n\R2rep\gamma\;\;=\;\;
\Dw\gamma\,\vd\cdot\n\R2rep\Dw\gamma \nonumber \\
&=&
\Dw\gamma\Rrep\vd\cdot\n\Rrep\Dw\gamma\;\;-\;\;
\Dw\gamma[\Rrep,\vd]\cdot\n\Rrep\Dw\gamma .
\end{eqnarray}
Integrating the first term on the right hand side
over $\RRn$ yields
\begin{eqnarray}
\int_{\RRn}
\Dw\gamma\Rrep\vd\cdot\n\Rrep\Dw\gamma & = &
\int_{\RRn}
\Rrep\Dw\gamma\,\vd\cdot\n\Rrep\Dw\gamma \nonumber \\
& = &
\int_{\RRn}\Big\{\frac{1}{2}\n\cdot\left
(\vd\mid\Rrep\Dw\gamma\mid^{2}\right)-
\frac{1}{2}
\n\cdot\vd\mid\Rrep\Dw\gamma\mid^{2}\Big\} \nonumber \\
& = &
-\frac{1}{2}\int_{\RRn}\n\cdot v_{\delta}|R_{\ep}\Dw\gamma|^{2}.
\end{eqnarray}
Treating this and the second term on the right hand side of
equation (2.23) as above
(cf. equations (2.13) and (2.15) respectively) we conclude
that
\begin{equation}
\lim_{\ep,\delta\rightarrow 0}\;\;\;\int_{\TTl}\int_{\RRn}
-\Dw\gamma\Dw\beta_{\ep,\delta}\;\;=\;\;
\frac{1}{2}\int_{\TTl}\int_{\RRn}
tr\,dv\,(\Dw\gamma)^{2}.
\end{equation}
\vspace{.2in}
Finally, we consider the nonlinear term
$f(\gamma)\beta_{\ep,\delta}$ in equation (2.8) which we write as
\begin{eqnarray}
f(\gamma)\beta_{\ep,\delta} & = &
f(\gamma)\vd\cdot\R2rep\n\gamma \nonumber \\
&=&
f(\gamma)\vd\cdot\n\gamma +
\Big[f(\gamma)\vd\cdot\R2rep\n\gamma -
f(\gamma)\vd\cdot\n\gamma\Big].
\end{eqnarray}
Since $\n\gamma$ exists (in the classical sense)
almost everywhere with
$\n\gamma f(\gamma)\in L^{1}(\RRn)$ and $F\in C^{1}(\RR,\RR)$,\ we
can write $\n F(\gamma) = \n\gamma f(\gamma)$. That is,
\ $F(\gamma)\in H^{1,1}(\RRn)$. Thus,
\begin{eqnarray}
\int_{\RRn}
f(\gamma)\vd\cdot\n\gamma & = &
\int_{\RRn}\vd\cdot\n F(\gamma) \nonumber \\
&=&
\int_{\RRn}\Big\{\n\cdot(\vd F(\gamma)) - \n\cdot\vd F(\gamma)
\Big\} \nonumber \\
& = &\int_{\RRn}
-\n\cdot\vd F(\gamma) \nonumber \\
& = &
\int_{\RRn}-\n\cdot\v F(\gamma)\;\; +
\;\;\int_{\RRn}(\n\cdot\vd - \n\cdot v) F(\gamma).
\end{eqnarray}
In going from the second to the third equality we have used the
fact that $\vd F(\gamma)\in (H^{1,1}(\RRn))^{N}$ (Lemma 2.8).
The last integrand on the right
tends to zero in $L^{1}(\RRn)$ as $\delta\rightarrow
0$ and by dominated convergence tends to zero in $L^{1}(\Omeg)$.
For the remaining terms on the right hand side of equation (2.26)
we have
\begin{equation}
\int_{\RRn}\Big[
f(\gamma) \vd\cdot\R2rep\n\gamma - f(\gamma) \vd\cdot\n\gamma
\Big]\;\;=
\;\;\int_{\RRn}
f(\gamma) \vd\cdot(\R2rep\n\gamma - \n\gamma).
\end{equation}
Because $\R2rep\n\gamma\rightarrow \n\gamma$ in $L^{2}(\RRn)$
with $\|\R2rep\n\gamma - \n\gamma\|_{2}\;\leq\;2\|\n\gamma\|_{2}$
and $f(\gamma)\in L^{2}(\RRn)$, the Schwarz inequality and the
boundedness of $\vd$ for $\delta>0$ imply that this
last integral converges
to zero as $\ep\rightarrow 0$ for almost all
$\theta$ and for each $\delta>0$, and hence by dominated convergence
converges to zero in $L^{1}(\TTl)$ as $\ep\rightarrow 0$ for each
$\delta >0$.
We conclude that
\begin{equation}
\int_{\TTl}\int_{\RRn}
f(\gamma)\beta_{\ep,\delta} =
-\int_{\TTl}\int_{\RRn}
tr\,dv\,F(\gamma) \;\;\;\;+ \;\;E_{1}(\ep,\delta)
\end{equation}
where $E_{1}(\ep,\delta)$ can be made arbitrarily small
first by choosing $\delta$ sufficiently small and then $\ep$
sufficiently small.
\vspace{.3in}
Combining all of the above results, we have shown that
\begin{eqnarray}
\lefteqn{\int_{\TTl}\int_{\RRn}
\Big\{-\Dw\gamma\Dw\beta_{\ep,\delta}
+ \n\gamma\cdot\nabla\beta_{\ep,\delta} +
f(\gamma)\beta_{\ep,\delta}\Big\}} \nonumber \\
& & =\;\; \int_{\TTl}\int_{\RRn}
\left\{tr\,dv\,\left(\frac{1}{2}
(\Dw\gamma)^{2}
- F(\gamma)\right) +
\nabla\gamma\cdot[dv - \frac{1}{2}tr\,dv{\bf 1}]\nabla\gamma
\rangle\right\} \nonumber \\
& & \;\;\;\;\;\;+\;\;E(\ep,\delta),
\end{eqnarray}
where $E(\ep,\delta)$ can be made arbitrarily small
by choosing $\delta$ and $\ep$ sufficiently small.
This, together with equation (2.8), completes the proof of Theorem
2.7
\ \ \ $\Box$
\vspace{.4in}
\section{Nonexistence of quasiperiodic solutions}
\setcounter{equation}{0}
\setcounter{cor}{0}
\setcounter{prop}{0}
The dilation group on $\RRn$ is the one-parameter group of
diffeomorphisms $\;\Phi_{\lambda}(x) = \lambda x,\;\;x\in\RRn,
\;\;\lambda
\geq 1$.
Its infinitesimal generator, the vector field
$v(x) = x$,
satisfies the hypothesis of Theorem 2.7. Here $dv = {\bf 1}$ and
$tr\,dv = N$. The following corollary then follows from
equation (2.4).
\begin{cor}
Let $v(x)=x$ be the generator
of dilations on $\RR^{N}$.
If $\gamma$ and $f$ satisfy the hypothesis of Theorem 2.7, then
\begin{equation}
\int_{\TTl}\int_{\RRn}\left\{\frac{1}{2}
(\Dw\gamma)^{2}
+ (\frac{2-N}{2N})\mid\nabla\gamma\mid^{2} -
F(\gamma)\right\} \;=\;0.
\end{equation}
\end{cor}
\noindent\underline{Remark:}\ \ In the case when $\gamma$ is
independent of $\theta$, so that $\gamma$ solves the nonlinear
elliptic equation $\Delta\gamma = f(\gamma)$, equation (3.1) is
the well known Pohozaev identity [Po].
The next proposition describes a class of virial relations that are
not derived from a vector field on $\RRn$.
\setcounter{prop}{1}
\begin{prop}
(Gauge transformations)\ Let
$\gamma\in H^{1}(\Omeg)$ satisfy (2.2).
If $h$ is a smooth function on $\RRn$
that is bounded along with its derivatives,
then
\begin{equation}
\int_{\TTl}\int_{\RRn}
\left\{h\left(-(\Dw\gamma)^{2}
\;+\;\mid\nabla\gamma\mid^{2} +
\;f(\gamma)\gamma\right)\;\;-\;\;
\frac{1}{2}(\Delta h)\gamma^{2}\right\}\;\;\;\;=\;0.
\end{equation}
\end{prop}
\noindent\underline{Proof}:
Since $h\gamma\in H^{1}(\Omeg)$, from equation (2.2)
\begin{equation}
\int_{\TTl}\int_{\RRn}
\Big\{-\Dw\gamma\Dw (h\gamma)\;+\;
\;\nabla\gamma\cdot\nabla(h\gamma) + f(\gamma)h\gamma\Big\}
\;\;\;=0.
\end{equation}
Expanding the second term in the integrand,
\begin{eqnarray}
\n\gamma\cdot\nabla(h\gamma) & = &
(\nabla\gamma\cdot\nabla h)\gamma + h\mid\nabla\gamma\mid^{2}
\nonumber \\
& = &
\frac{1}{2}\nabla\cdot(\gamma^{2}\nabla h)
+ h\mid\nabla\gamma\mid^{2}
- \frac{1}{2}(\Delta h)\gamma^{2}.
\nonumber
\end{eqnarray}
The first term on the right vanishes when integrated over $\RRn$ and so
equation (3.2) follows\ \ \ $\Box$
\vspace{.2in}
By combining the previous two virial relations, take
$h\equiv c$ in equation (3.2) and add this to equation (3.1), we
obtain the following
identity that allows us to prove several nonexistence theorems for
quasiperiodic solutions. A special case of this identity was obtained
in [V1] for periodic solutions on $\RR^{1}$.
\setcounter{prop}{2}
\begin{prop}
Let $\gamma$ and $f$ satisfy the hypothesis of Theorem 2.7.
Then for any $c\in\RR$,
\begin{equation}
\int_{\TTl}\int_{\RRn}\left\{(\frac{1}{2}-c)(\Dw\gamma)
^{2} +
(c + \frac{2-N}{2N})\mid\nabla\gamma\mid^{2}\right\}\;\;=\;\;
\int_{\TTl}\int_{\RRn}
\left\{F(\gamma) - c\gamma f(\gamma)\right\}.
\end{equation}
\end{prop}
\vspace{.2in}
\begin{prop}
Let $\gamma$ be a weak solution of NLW on $\Omeg$ with $\gamma$
and $f$ satisfying the hypothesis of Theorem 2.7.
If for some $c\in [\frac{N-2}{2N},\,\frac{1}{2}]$,
\ $F(z) - czf(z)\;\leq\;0$ for all $ z$
such that
$\mid z\mid\leq \|\gamma\|_{L^{\infty}(\Omeg)}$,
then $\gamma$ is independent of $\theta$.
\end{prop}
\noindent\underline{Proof}:
By hypothesis the left hand side of equation (3.4)
is nonnegative while the right hand side
is nonpositive. Hence, both sides must be zero. For $c\in(
\frac{N-2}{2N},\frac{1}{2}]$
this implies that $\int_{\TTl}\|\n\gamma\|_{2}^{2} = 0$
so that
for almost all $\theta,\;\;\n\gamma = 0$ almost everywhere on $\RRn$.
That is, $\gamma(\theta)$ is constant almost everywhere on
$\RRn$ for almost all $\theta$. Because $\gamma$ is continuous,
\ $\gamma$ is therefore constant on $\Omeg$.
Since $\gamma\in L^{2}(\Omeg)$, \ $\gamma(\theta)\in L^{2}(\RRn)$
for almost all $\theta$ so that this constant must be zero.
In the case $c=\frac{N-2}{2N}$ equation (3.4)
implies that $\Dw\gamma =
0$ almost everywhere on $\Omeg$.
This implies that for almost all
$x\in\RRn,\;\;\gamma$ is invariant under the flow
$\theta\mapsto\theta + \omega t$. Since this flow is ergodic ($\omega$
is incommensurate),
$\gamma$
is constant on $\TTl$ for these $x$ ([Pe], Prop 2.4.1).
Therefore, $\gamma$ is independent of $\theta$ almost everywhere
on $\RRn$
\ \ \ $\Box$
\vspace{.2in}
\noindent\underline{Remark:}\ \ Let ${\cal Z} = \{z\in\RR\;;\;\;
F(z) - czf(z) = 0\}$. Then, under the hypothesis that
$F(z) - czf(z) \leq 0$ for all $z$ such that $|z|\leq
\|\gamma\|_{L^{\infty}
(\Omeg)}$,
\ \ $\int_{\Omeg}\{F(\gamma)
- c\gamma f(\gamma)\} = 0$ implies that $\gamma(x,\theta)\in
{\cal Z}$ for all $(x,\theta)\in\Omeg$. If ${\cal Z}$ is
composed of isolated points then, since $F(\gamma),\;f(\gamma)$
and $\gamma$ are continuous,
$\gamma$ must be constant on $\Omeg$. If ${\cal Z}$ contains an
interval then this argument does not imply that $\gamma$
is constant.
\vspace{.2in}
If $\varphi$ is a weak quasiperiodic
solution of NLW (Definition 2.6), then applying
Proposition 3.4 to its generating function we obtain the
following theorem concerning the nonexistence of quasiperiodic
solutions.
\setcounter{thm}{4}
\begin{thm}
(Nonexistence of quasiperiodic solutions of NLW)\
\ Suppose
$\varphi$ is a weak $l$-quasiperiodic solution of NLW
on $\RR^{N+1}$
with frequency $\omega$.
Let $\gamma$ be the generating function of $\varphi$ and assume
that $\gamma$
and $f$ satisfy the hypothesis of Theorem 2.7.
If for some $c\in[\frac{N-2}{2N},\,\frac{1}{2}]$,
\begin{equation}
F(z) - czf(z)\leq 0 \mbox{ for all } z \mbox{ such that }
\mid\! z\!\mid\leq
\|\varphi\|_{L^{\infty}(\RR^{n+1})}
\end{equation}
where $F(z)\equiv\int_{0}^{z}f(w)\,dw$, then
$\varphi$ is independent
of time.
\end{thm}
\noindent \underline{Remarks:} \nobreak
We point out that condition (3.5) is compatible with our
hypothesis (H1) on the nonlinearity. For instance, if we are
considering bounded solutions
then any polynomial is an admissible nonlinearity (i.e., bounded
solutions and polynomial nonlinearities satisfy
the hypothesis of Theorem 2.7), while in
Example 1 below we exhibit a polynomial satisfying (3.5).
Sufficient (but not necessary) conditions on the nonlinearity such
that it satisfies the inequality
$F(z) - czf(z)\leq 0$ are as follows.
Suppose $F(z)=H(z^{a})$ for some
convex function $H$. Then
\begin{displaymath}
F'(z) = f(z) = az^{a-1}H'(z^{a}).
\end{displaymath}
Since $H$ is convex, \ $H'(w)\leq H'(y)$ when $w\leq y$.
Integrating this inequality with respect to $w$ from 0 to $y$ gives
$H(y)\leq yH'(y)$. Setting $y=z^{a}$;
\ $F(z)-z^{a}H'(z^{a})\leq 0$,
and hence\ $F(z)-a^{-1}zf(z)\leq 0$.
In particular, for a pure power law nonlinearity $f(z)= az^{\alpha}$,
then $F(z) - \frac{1}{\alpha + 1}zf(z) = 0$.
The condition $\frac{1}{\alpha + 1}\in [\frac{N-2}{2N},\frac{1}{2}]$
implies then that NLW with this nonlinearity has no quasiperiodic
solutions for any $\alpha\geq 1$ in the case $N=1,2$ or for
$1\leq\alpha\leq\frac{N+2}{N-2}$ in the case $N>2$.
The proof of Theorem 3.5
makes no use of the frequency
$\omega$ of the solution:\ If
the nonlinearity satisfies condition
(3.5) then NLW will not supporÔˆÿt any kind of quasiperiodic
solution, where by "any kind" we mean ÿ
quasiperiodic functions with any frequency.
>From the point of view of virial relations
the nonlinearity does not distinguish between
different frequencies or
dimensions in the time variable of quasiperiodic
solutions (see also
the remark following Proposition 4.4 below).
\vspace{.4in}
\noindent\underline{ {\bf Examples and Applications}}
\nobreak
\vspace{.3in}
\noindent 1)\ \ \ \ Let
$f(z)
= a_{1}z + a_{3}z^{3} + \cdots + a_{2m+1}z^{2m+1}$ be a polynomial
with only odd powers of $z$. Then
\begin{displaymath}
F(z) - czf(z) = a_{1}(\frac{1}{2}-c)z^{2} +
a_{3}(\frac{1}{4} - c)z^{4} + \cdots +
a_{2m+1}(\frac{1}{2m+2}-c)z^{2m+2}.
\end{displaymath}
Here $\mid\! f(z)\!\mid \;\leq\; c_{1}\,(\mid\! z\!\mid +
\mid\! z \!\mid^{2m+2}
)$ so that, referring
to the hypothesis of Theorem 2.7, the generating
functions that are covered in our analysis for this particular
nonlinearity are those that are of class $L^{4(m+1)}(\Omeg)$
if $N+l>2$. If $N+l=2$ (periodic solutions on $\RR$)
any generating function is admissible.
(Recall that our
definition of the generating function included the requirement
that it be of class
$H^{1}(\Omeg)$.)
If $a_{2k+1}\geq 0,\;k=0,\ldots,m$,\ then
by choosing $c=1/2$ we conclude,
using Theorem 3.5, that
NLW with
this nonlinearity has no weak $l$-quasiperiodic solution,
for any $l$, whose generating function is of class
$L^{4(m+1)}(\Omeg)$
in spatial dimension $N>1$, or whose generating function is
of class $H^{1}(\Omega_{1,1})$ if $N+l=2$.
In particular, generating functions that
are bounded satisfy these criteria.
\noindent\underline{Remarks:}
Other possibilities for the choice of $c$ to derive necessary
conditions for existence
may arise if the coefficients are not
all positive. For example, if $a_{1}\leq 0$ and
$a_{2k+1}\geq 0$,\ $k=1,\ldots,m$, then we reach the same
conclusion with any
$c\in [1/4,\,1/2]$, which is a valid interval for $c$ if
$N\leq 4$ (that is, we can apply Theorem 3.5).
Global (in time)
existence for some equations of this form is proven in
[GV] (see also [Str2]).
The class of solutions considered there is somewhat different than
that considered here but our results suggest that these equations
possess no quasiperiodic solutions even though the Cauchy problem
leads to global solutions.
In Section 5 we will extend Theorem 3.5 to
almost periodic solutions (Theorem 5.1) so that in fact
these equations may not have
almost periodic solutions either.
Consequently, these nonlinear wave equations may have no
bound states (cf. the discussion in the introduction).
\vspace{.3in}
\noindent 2)\ \ \ \ \underline{Small Amplitude Solutions}
\nobreak
If a solution has small amplitude, i.e., if it has small $L^{\infty}
(\RR^{N+1})$
norm, then only the properties of $f$ in a neighborhood of the
origin contribute to the dynamics.
In particular, referring to equation (3.5), if
$F(z) - czf(z)\leq 0$ in a neighborhood of zero for some $c\in
[\frac{N-2}{2N},\frac{1}{2}]$, then NLW has no
small amplitude quasiperiodic solutions.
A particular case where small amplitude solutions arise is when
NLW has a
family of localized, periodic solutions
( called "breathers")
that originate from the zero solution (see, for example, [BMW],[SK]).
An example of such a family is provided by the
sine-Gordon equation on $\RR^{1+1}$ (recall equation (1.3)).
We now present two nonexistence results for small
amplitude quasiperiodic solutions based on Theorem 3.5 by using
a more detailed description of the nonlinearity.
Here we assume that $f$ can be expanded in a Taylor series about the
origin as
\begin{equation}
f(z) \;=\;f'(0)z + \frac{f^{(2k+1)}(0)}{(2k+1)!}z^{2k+1}
+ R(z)
\end{equation}
where $f^{(2k+1)}(0)\neq 0$,\ $k\geq 1$ and $R(z) =
O(|z|^{2k+2})$.
To state the next two corollaries we will require the following
definition.
\vspace{.1in}
\noindent{\bf Condition A}
\ \ With reference to (3.6), the three numbers
$(f^{(2k+1)}(0),k,N)$ satisfy the following conditions ($N$ is the
spatial dimension).
If $f^{(2k+1)}(0)<0$,
then $N\leq 3$;\ in the case $N=1$ or $2$, $k$
can be any positive integer, in the case $N=3$,\ $k=1$.
If $f^{(2k+1)}(0)>0$ then $k$ and $N$ can be any positive integers.
\setcounter{cor}{5}
\begin{cor}
Let $f$ have a Taylor series at the origin of the form (3.6).
If either
\ (i) $f'(0)<0$, or \ (ii) $f'(0)>0$ and $f^{(2k+1)}(0)>0$, or
\ (iii) $f'(0)=0$
with $(f^{(2k+1)}(0), k, N)$ satisfying Condition A, then
NLW has no weak $l$-quasiperiodic
solutions $\varphi$
of sufficiently small amplitude
for any $l\in\NN$ and for any frequency $\omega\in\RR^{l}$.
This result holds in any spatial dimension in the cases
(i) and (ii) and in those spatial dimensions determined by
Condition A in the case (iii).
\end{cor}
\noindent
\underline{Proof}:
We work with the generating function $\gamma$ of $\varphi$.
Since $\varphi$, and hence $\gamma$,
is of small amplitude it is, in particular, bounded.
Therefore, as we will apply
Proposition 3.3 it is enough that
$f$ satisfy a Lipschitz condition at
the origin (cf. statement (ii) in Theorem 2.7 with
$q=\infty$), which it clearly does.
We have that
\begin{equation}
F(z) - czf(z) = f'(0)(\frac{1}{2}-c)z^{2} +
\frac{f^{(2k+1)}(0)}{(2k+1)!}\left(\frac{1}{2k+2} -
c\right)z^{2k+2} +
\tilde{R}(z)
\end{equation}
where $\tilde{R}(z) = \int_{0}^{z}R(w)\,dw - czR(z)$.
We have expanded the integrand on the right hand side of (3.4).
Basing our analysis on (3.7),
our goal is to adjust the parameter $c$, within the interval
$[\frac{N-2}{2N},\frac{1}{2}]$, according to the
properties of the Talor series of $f$ so that the right hand side of
(3.4) is strictly negative for sufficiently small solutions
$\gamma$, $\gamma\neq 0$. Then, because the left hand side of (3.4)
will be
nonnegative, this contradiction will imply that $\gamma$ must in
fact be zero.
We first consider the case $f'(0)<0$. Thus, the lowest order term
on the right hand side of (3.7) is negative. In this case we
set $c=\frac{1}{2k+2}$ (any $c\in[\frac{N-2}{2N},\frac{1}{2})$ will
do, though).
If $f'(0)>0$ then we set $c=\frac{1}{2}$.
Then since $\frac{1}{2k+2}-\frac{1}{2}<0$, if
$f^{(2k+1)}(0)>0$, regardless of $k$, the lowest order term
on the right hand side of (3.7) will be negative.
If $f'(0)=0$, then if $f^{(2k+1)}(0)<0$ we set $c=\frac{N-2}{2N}$
which is less that $\frac{1}{2k+2}$ for those $k$ and $N$ specified
in Condition A. If $f^{(2k+1)}(0)>0$ we set $c=\frac{1}{2}$. In
either case the second term on the right hand side of (3.7) is
negative for $z\neq 0$.
With $c$ specified in this way
the lowest order term on the right hand
side of (3.7) is strictly negative for $z\neq 0$.
Now we show that the remainder term $\tilde{R}(z)$ does not upset
this for $z$ sufficiently small.
Applying the mean value theorem to
the remainder term in the Taylor series of $F$, we see that
$$
\tilde{R}(z)\;=\;\frac{f^{(2k+2)}(u)}
{(2k+3)!}z^{2k+3}
$$
for some $u = u(z),$\ $|u|< |z|$.
Therefore,
in the case when $c=\frac{1}{2k+2}$ \ there is a $d_{1}>0$ such that
for $|z|\leq d_{1}$, $z\neq 0$,
\begin{displaymath}
\mid\tilde{R}(z)\mid\;\;<\;\;\left|
f'(0)(\frac{1}{2} - \frac{1}{2k+2})\right|z^{2}.
\end{displaymath}
In the case when $c=\frac{1}{2}$ or $c=\frac{N-2}{2N}$,
there exists a $d_{2} > 0$ (which depends on $c$)
such that for $\mid\! z\!\mid \leq d_{2}$, $z\neq 0$,
\begin{displaymath}
\mid\tilde{R}(z)\mid\;\;<\;\;\left|
\frac{f^{(2k+1)}(0)}{(2k+1)!}\left(\frac{1}{2k+2} - c
\right)\right|z^{2k+2}.
\end{displaymath}
Therefore, for $|z|
\leq d = \min\{d_{1},d_{2}\}$, $z\neq 0$,
and with $c$ specified as above,
the right hand side
of equation (3.7) is strictly negative.
This implies that
if $\gamma$ is a nonzero solution of NLW on $\Omeg$ such that
$\|\gamma\|_{L^{\infty}(\Omeg)}\leq d$,
then for this $c$ the right hand side of (3.4) is strictly
negative.
However the left hand side of (3.4) is nonnegative.
Therefore, there cannot be such a solution
\ \ \ $\Box$
\vspace{.2in}
\noindent\underline{Remark:}
\ \ If one is considering nonlinear Klein-Gordon equations
(i.e., $f'(0)>0$) with odd nonlinearity, which is typical in
physical applications, then by statement (ii) of the corollary a
necessary condition for the existence of small amplitude
quasiperiodic solutions in any spatial dimension
is that the next highest term in the Taylor
series after the linear term have a negative coefficient.
\vspace{.2in}
We now demonstrate how a priori information about the solution can be
used in conjunction with virial relations to derive a
nonexistence result for periodic solutions on $\RRn$ (here
the generating function $\gamma$ of $\varphi$ is just
$\gamma(x,t) = \varphi(x,t/\omega)$).
We will obtain a result similar to Coron's [Co]
in multi-spatial dimensions, but only
for small amplitude periodic solutions with zero mean.
We here that $f'(0)>0$, the case $f'(0)\leq 0$
would be covered in a way analogous to how it was treated in
the previous corollary.
Referring to (3.7),
the virial relation of Proposition 3.3 (equation (3.4))
can be written as
\begin{eqnarray}
\lefteqn{(\frac{1}{2} - c)\Big(\|\Dw\gamma\|^{2}_{L^{2}
(\Omega_{N,1})}
\;-\;f'(0)\|\gamma\|^{2}_{L^{2}(\Omega_{N,1})}\Big)
\;\;+\;\;(c+\frac{2-N}{2N})\|\n\gamma\|
^{2}_{L^{2}(\Omega_{N,1})}}
\nonumber \\
& &
\;\;\;\;\;\;=\;\;\int_{\TT^{1}}\int_{\RRn}
\left\{\frac{f^{(2k+1)}(0)}{(2k+1)!}\left(\frac{1}{2k+2} -c\right)
\gamma^{2k+2}
\;+\; \tilde{R}(\gamma)\right\}.
\end{eqnarray}
>From this we see that if we had some a priori estimate on how
$\|\Dw\gamma\|^{2}_{L^{2}(\Omega_{N,1})}$ compares to $\|\gamma\|^{2}
_{L^{2}(\Omega_{N,1})}$ \ so as to make definite the sign of
$(\|\Dw\gamma\|^{2}_{L^{2}(\Omega_{N,1})} - f'(0)\|\gamma\|^{2}
_{L^{2}(\Omega_{N,1})})$, then we would not be forced to set $c = 1/2$
to obtain a necessary condition for existence (as in the previous
corollary).
An instance of when this is possible is provided by Wirtinger's
inequality for periodic functions along with an additional
hypothesis on the frequency of $\gamma$.
We first recall this inequality.
\ $\gamma$ has a Fourier expansion
\ $\gamma = \sum_{k} \gamma_{k}(x)e^{ikt}$ (we are writing $t$
instead of $\theta$).
Then, since $\gamma\in H^{1}(\Omega_{N,1})$,
\ $\partial_{t}\gamma$ has the Fourier expansion
$\partial_{t}\gamma = \sum_{k} ik\gamma_{k}(x)e^{ikt}$.
Therefore,
\ $\|\partial_{t}\gamma\|^{2}_{L^{2}
(\Omega_{N,1})}\;=\;\sum_{k} k^{2}\|\gamma_{k}\|^{2}_{L^{2}(\RRn)}$.
If in addition $\gamma_{0} = 0$ (i.e., if $\varphi$ has zero
mean) we derive the inequality
$\|\partial_{t}\gamma\|^{2}_{L^{2}(\Omega_{N,1})}\;\geq
\;\|\gamma\|^{2}_
{L^{2}(\Omega_{N,1})}$, which implies that
$\|\Dw\gamma\|^{2}_{L^{2}(\Omega_{N,1})} = \omega^{2}\|\partial_{t}
\gamma\|^{2}_{L^{2}(\Omega_{N,1})} \geq \omega^{2}
\|\gamma\|^{2}_{L^{2}(\Omega_{N,1})}$ (recall that
$\Dw = \omega\partial_{t}$ in this case).
Substituting this inequality into (3.8), if $\omega^{2}>f'(0)$
then $(\|\Dw\gamma\|^{2}_{L^{2}(\Omega_{N,1})} - f'(0)
\|\gamma\|^{2}_{L^{2}(\Omega_{N,1})})\geq 0$ with equality holding
if and only if $\gamma = 0$.
Thus, if $\omega^{2}> f'(0)$
the left hand side of (3.8) is nonnegative for
$c\in[\frac{N-2}{2N},\frac{1}{2}]$.
Actually, the left hand side of (3.8) is strictly positive for
$c\in[\frac{N-2}{2N},\frac{1}{2}]$ and $\gamma\neq 0$;
\ that this is true
when $c\in[\frac{N-2}{2N},\frac{1}{2})$ was just pointed out,
while if $c=\frac{1}{2}$, then $\|\n\gamma\|_{L^{2}(\Omeg)}^{2}
=0$ implies that $\gamma=0$, as explained in the proof
of Proposition 3.4.
Now we determine
conditions under which the right hand side of (3.8) is
nonpositive.
If these conditions are met, then the assumptions
$\omega^{2} > f'(0)$ and $\varphi$ having zero mean, together which
we have just seen leads to the conclusion that the
left hand side of (3.8) is strictly positive if
$\varphi\neq 0$, will imply that $\varphi$ must be zero.
First note that, because $\tilde{R}(\gamma) = O(|\gamma|^{2k+3})$, if
$\gamma$ is of sufficiently small amplitude
the right hand side of (3.8) will be dominated by the first term
in the integrand (as described in the previous corollary).
We choose $c$ so as to make $f^{(2k+1)}(0)\Big(
\frac{1}{2k+2} - c\Big) < 0$. If $f^{(2k+1)}(0)>0$ then set
$c=\frac{1}{2}$.
If $f^{(2k+1)}(0)<0$ then set $c=\frac{N-2}{2N}$.
In either case the left hand side of (3.8) is strictly positive
if $\gamma\neq 0$, while
if $(f^{(2k+1)}(0),k,N)$ satisfies Condition A, the right hand side
of (3.8) is nonpositive for sufficiently small $\gamma$.
Therefore, there cannot be such solutions $\gamma$.
This completes the proof of the following
corollary.
\setcounter{cor}{6}
\begin{cor}
Let $f$ have a Taylor series at zero of the form (3.6). Assume that
$f'(0)>0$ and that $(f^{(2k+1)}(0),k,N)$ satisfies Condition A.
Let $\varphi$ be a small amplitude weak $2\pi/\omega$ -periodic
solution of NLW on $\RR^{N+1}$.
In addition suppose that $\varphi$ has zero mean, i.e.,
\newline
$\int_{0}^{2\pi/\omega} \varphi(x,t)\,dt\;=\;0$ for all $x\in\RRn$.
Then $\omega^{2}\leq f'(0)$.
\end{cor}
\noindent\underline{Remarks:}
The proof of this corollary was carried out by contradiction
and was
outlined above.
We required $\varphi$ to have zero mean so as to be able
to use Wirtinger's inequality, which provided us with an a priori
estimate.
By a small amplitude solution we mean that $\varphi$ has sufficiently
small $L^{\infty}(\RR^{N+1})$ norm, sufficiency being determined by
the values of $f'(0),f^{(2k+1)}(0)$ and the parameter $c$ whose
value was assigned
during the proof above. To illustrate this precisely let
$\{\ep_{n}\}$
be a sequence of positive numbers converging to zero and suppose
that for each $n_{o}\in\NN$ there is an $\bar{n}\geq n_{o}$
such that there exists a (nontrivial) $2\pi/\omega$-periodic
solution $\varphi_{\bar{n}}$ of NLW with zero mean with
$\|\varphi_{\bar{n}}\|_{L^{\infty}(\RR^{N+1})}\leq\ep_{\bar{n}}$.
If $f'(0)>0$ and
if ($f^{(2k+1)}(0),k,N)$ satisfies
Condition A, then
$\omega^{2}\leq f'(0)$.
One may try to
prove a similar result for small amplitude
{\em quasiperiodic} breathers. However, a Wirtinger-type
inequality for quasiperiodic functions is generally not possible
for the following reason. Let $\gamma\in H^{1}(\Omeg)$, $l>1$,
be the generating function
of $\varphi$ with Fourier series $\sum_{k\in\ZZ^{l}}\gamma_{k}(x)
e^{ik\cdot\theta}$ (so that $\varphi (x,t) = \sum_{k\in\ZZ^{l}}
\gamma_{k}(x)e^{ik\cdot\omega t}$).
Then, $\|\gamma\|
_{L^{2}(\Omeg)}^{2}\;=\;\sum_{k\in\ZZ^{l}}\|\gamma_{k}\|_{2}^{2}$
\ and $\|\Dw\gamma\|^{2}_{L^{2}(\Omeg)} =
\sum_{k\in\ZZ^{l}}\mid\! k\cdot\omega\!\mid^{2}
\|\gamma_{k}\|^{2}_{2}$.
Suppose that $\gamma_{0} = 0$. Because $\omega$ is incommensurate,
\ $k\cdot\omega$ becomes arbitrarily small infinitely often as
$k$ ranges over $\ZZ^{l}$. Thus it is impossible to
bound $\mid\!k\cdot\omega\!\mid^{2}$ from below by a
strictly positive number $c$ and obtain an
inequality of the form
\ $\|\Dw\gamma\|_{L^{2}(\Omeg)}^{2} \geq c\|\gamma\|_{L^{2}
(\Omeg)}^{2}$.
\vspace{.4in}
\noindent 3)\ \ \underline
{Local vector fields}
The preceding
examples assumed that the solutions were of small amplitude.
Since solutions are a priori in $H^{1}(\Omeg)$, they
decay in an average sense as $\mid\!x\!\mid\rightarrow\infty$.
Making the additional assumption that this decay is pointwise
and uniform in $t$, we could obtain some of
the previous results without
any assumptions of small amplitude provided that the virial relation
was localized in a neighborhood of infinity.
We illustrate this idea with an example concerning quasiperiodic
solutions on $\RR^{1+1}$.
\setcounter{cor}{7}
\begin{cor}
Suppose
$\varphi$ is a weak $l$-quasiperiodic solution of NLW on $\RR^{1+1}$
with frequency $\omega$
that converges to zero as $|x|\rightarrow
\infty$ uniformly in $t$.
If $F(z)\leq 0$ in a neighborhood of
zero, then \ $\varphi\equiv 0$.
\end{cor}
\noindent\underline{Remark:}\ \ If $F(z)\leq 0$ in a neighborhood
of zero, then this implies that $f'(0)\leq 0$.
\vspace{.1in}
\noindent\underline{Proof:}
Let $\rho>1$ and define $g: \RR\rightarrow\RR$ by
\vspace{.2in}
$\begin{array}{lllrcl}
g(x) &=& 6x^{2} + (4-12\rho)x + (6\rho^{2} - 4\rho + 1)
& \rho \leq& x & \\
&=& (x-\rho + 1)^{4} & \rho -1\leq & x & \leq\rho \\
&=& 0 & -\rho + 1\leq& x & \leq\rho -1 \\
&=& (x + \rho -1)^{4}&
-\rho\leq & x & \leq -\rho + 1 \\
&=& 6x^{2} -
(4 - 12\rho)x + (6\rho^{2} - 4\rho + 1)
& & x & \leq -\rho
\end{array}$\newline
On $\RR$ define the vector field $v(x) = g'(x)$.
Then $v$ satisfies the hypothesis of Theorem 2.8 except that
$\Delta v$ has a finite
discontinuity at $\mid\! x\!\mid = \rho.$
This does not affect the
validity of the theorem, however
(see equations (2.15) and (2.23):\ this
is the only place in the proof where derivatives
of $v$ of order greater than one are encountered).
Here we have $v = 0$ on $\{\mid\! x\!\mid\leq\rho-1\}$
with $dv = g'' \geq 0$. For this vector field equation (2.7) reads
\begin{equation}
\int_{\TTl}\int_{\mid x\mid>\rho -1}
dv\left\{\frac{1}{2}(\Dw\gamma)^{2}
+ \frac{1}{2}\mid\n\gamma\mid^{2} - F(\gamma)\right\}\;\;=\;\;0.
\end{equation}
If $F(z)\leq 0$ in a neighborhood of zero then, by taking
$\rho$ sufficiently large, equation (3.9) implies
that $\gamma =0$ on the support of $v$.
The conclusion of the corollary then follows from the next
Proposition, the proof of which we leave as an open problem (see,
however, Theorem 5.5 of [PS]
which states that periodic
solutions with compact spatial support are identically zero -
we expect the same to be true for quasiperiodic solutions).
\setcounter{prop}{8}
\begin{prop}
Suppose $\gamma$ satisfies (2.2) and vanishes in a
neighborhood of infinity on $\RRn$ uniformly in $\theta$.
Then $\gamma\equiv 0$.
\end{prop}
\section{Virial relations for almost periodic solutions}
\setcounter{thm}{0}
\setcounter{prop}{0}
\setcounter{cor}{0}
\setcounter{defn}{0}
\setcounter{equation}{0}
\subsection{Almost periodic solutions}
To motivate our approach to almost periodic solutions we return
for a moment to the quasiperiodic case. Recall that our definition of
a weak solution $\gamma$ of NLW on $\Omeg$ with frequency
$\omega$ was that $\gamma\in H^{1}
(\Omeg)$ and for all $\beta\in H^{1}(\Omeg)$,
\begin{equation}
\int_{\TTl}\int_{\RRn}
\Big\{-\Dw\gamma\Dw\beta + \n\gamma\cdot\n\beta + f(\gamma)\beta
\Big\} = 0.
\end{equation}
We set
\begin{displaymath}
h(\theta)\;\;\;=\;\;\;
\int_{\RRn}
\Big\{-\Dw\gamma\Dw\beta + \n\gamma\cdot\n\beta + f(\gamma)
\beta\Big\}\;\;\;\;\in L^{1}(\TTl).
\end{displaymath}
Because $\omega$ is
incommensurate the flow
$\theta\mapsto\theta + \omega t$ on $\TTl$ is ergodic.
It follows, therefore, by a
result from ergodic theory (see for example [Pe] Thm. 2.2.3)
that
\begin{equation}
(2\pi)^{-l}\int_{\TTl} h(\theta)\,d\theta\;\;=
\;\;\ergint h(\omega t)\,dt.
\end{equation}
Defining
$\varphi(t)\equiv\gamma(\omega t)$ and $\psi(t)\equiv\beta(\omega t)$
we have that $\Dw\gamma(\omega t)\Dw\beta(\omega t) =
\partial_{t}\varphi(t)\partial_{t}\psi(t)$.
Combining equations (4.1) and (4.2) we obtain
\begin{equation}
\ergint\int_{\RRn}
\Big\{-\partial_{t}\varphi\partial_{t}\psi + \n\varphi\cdot\n\psi +
f(\varphi)\psi \Big\}= 0.
\end{equation}
Similarly, we
can also cast the virial relation of Theorem 2.7, equation (2.4),
in terms of $\varphi$ ;
\begin{equation}
\ergint\int_{\RR^{N}}
\left\{tr\,dv\; \big(\frac{1}{2}(\partial_{t}\varphi)^{2}
- F(\varphi)\big)\;+\;
\;\nabla\varphi\cdot\big[dv-\frac{1}{2}tr\,dv{\bf 1}\big]\,\nabla\varphi
\right\}\;\;=\;\;0.
\end{equation}
The property that $\ergint h(t)$ exists is shared by functions
from a larger class than the
quasiperiodic functions\ -\ the
almost periodic functions (for an introduction
to almost periodic functions see [C] or [LZ]).
We will see that formula (4.4) holds for solutions from this
larger class.
For any Banach space ${\cal B}$ we denote by
${\cal AP}(\RR,{\cal B})$ the Banach space of ${\cal B}$ valued
almost periodic functions on $\RR$. ${\cal AP}(\RR,{\cal B})$ is a
closed subspace of the space of bounded continuous functions
from $\RR$ to ${\cal B}$ with the uniform norm $\|h\| =
\sup_{t\in\RR}\|h(t)\|_{{\cal B}}$.
The class of almost periodic solutions we consider
is described in the following definition.
\begin{defn}
For any Banach space ${\cal B}$ let ${\cal AP}(\RR,{\cal B})$ denote
the space of ${\cal B}$ valued almost periodic functions on $\RR$.
Let ${\cal AP} \equiv
\{\varphi\in
{\cal AP}(\RR,H^{1}(\RRn))$
such that $\partial_{t}\varphi$
exists in the
strong sense as a uniformly continuous map
$\RR\rightarrow L^{2}(\RRn)\}$, and for
$q\in[2,\infty]$ let
${\cal AP}_{q} \equiv \{\varphi\in
{\cal AP}(\RR,H^{1}(\RRn)\cap L^{q}(\RRn))$
such that $\partial_{t}\varphi$
exists in the
strong sense as a uniformly continuous map
$\RR\rightarrow L^{2}(\RRn)\}$.
\end{defn}
That $\partial_{t}\varphi$ is uniformly
continuous as indicated
implies that $\partial_{t}\varphi\in\apll$ ([LZ] pp3).
Equation (4.3) motivates our definition of weak almost periodic
solution to NLW, but first we state what we mean by a classical
almost periodic solution.
\begin{defn}
$\varphi: \RR^{N+1}\rightarrow\RR$ is a {\em classical almost periodic
solution of NLW} if $\varphi$ solves
NLW and such that $\partial_{t}^{2}\varphi\in
L^{2}_{loc}(\RR,L^{2}(\RRn))$,
$\varphi\in C^{2}(\RR^{N+1})\cap{\cal AP}_{q}$
for some $q\in[2,\infty]$.
\end{defn}
\begin{defn} \ \ $\varphi: \RR^{N+1}\rightarrow\RR$
is a {\em weak almost periodic
solution of NLW} if $\varphi\in{\cal AP}$
and
\begin{equation}
\ergint\int_{\RR^{N}}
\Big\{-\partial_{t}\varphi\partial_{t}\psi +
\nabla\varphi\cdot\nabla\psi + f(\varphi)\psi\Big\}
\;\;=\;\;0\;\;\;\;\;\;\forall\;\psi\in{\cal AP}.
\end{equation}
\end{defn}
Our definitions of weak and strong
almost periodic solutions are compatible, that is, a strong
almost periodic solution is a weak almost periodic solution.
As in the quasiperiodic case, showing this entails integrating
by parts and consequently the issue of whether the boundary
terms vanish arises.
This is where the condition $\partial_{t}^{2}\varphi\in
L^{2}_{loc}(\RR,L^{2}(\RRn))$ enters, as we will see presently.
Note that if $\varphi$ is a
quasiperiodic function with generating function $\gamma$, then
$\gamma\in H^{2}(\Omeg)\Rightarrow\partial_{t}^{2}\varphi\in
L_{loc}^{2}(\RR,L^{2}(\RRn))$ (cf. the paragraph following
Definition 2.6).
\setcounter{prop}{3}
\begin{prop}
If $\varphi$ is a classical almost periodic solution of NLW then
$\varphi$ is a weak almost periodic solution of NLW.
\end{prop}
\noindent\underline{Proof}:
Since $\varphi$ is a solution of NLW,
for any $\psi\in{\cal AP}$ and any $T > 0$,
\begin{displaymath}
\frac{1}{T}\int_{0}^{T}\int_{\RRn}
\left(\partial^{2}_{t}\varphi - \Delta\varphi + f(\varphi)\right)
\psi\;\;\;= \;\;0.
\end{displaymath}
Interchanging the order of integration for the first term (
we can use Fubini's
Theorem since $\|(\partial_{t}^{2}\varphi)\psi\|
_{L^{1}(\RRn)}\in L_{loc}^{1}(\RR)$
),
\begin{displaymath}
\frac{1}{T}\int_{\RRn}\int_{0}^{T}(\partial^{2}_{t}\varphi)\psi =
\frac{1}{T}\int_{\RRn}\int_{0}^{T}\left\{
\partial_{t}(\partial_{t}\varphi\,\psi) - \partial_{t}
\varphi\partial_{t}
\psi\right\}.
\end{displaymath}
Since $(\int_{\RRn}(\partial_{t}\varphi)\psi )|^{T}_{0}$ is bounded
independently of $T$\ (
$\partial_{t}\varphi$ and $\psi$ are both in
$L^{\infty}(\RR,L^{2}(\RRn))$ so that
$(\partial_{t}\varphi)\psi\in L^{\infty}
(\RR,L^{1}(\RRn)))$, \ $\lim_{T\rightarrow\infty}\frac{1}{T}
\int_{\RRn}\int_{0}^{T}\partial_{t}
(\partial_{t}\varphi\psi)\;=\;0$.
Integrating the next term by parts,
\begin{displaymath}
\int_{\RRn}-\Delta\varphi\psi = \int_{\RRn}\n\varphi\cdot\n\psi,
\end{displaymath}
we conclude that
\begin{displaymath}
\ergint\int_{\RRn}
\left\{-\partial_{t}\varphi\partial_{t}\psi +
\n\varphi\cdot\n\psi + f(\varphi)\psi\right\} = 0
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\Box
\end{displaymath}
\noindent\underline{Remark}:
In the references [C] and [LZ], $\apb$ is defined as the uniform
closure
of the set of trigonometric
polynomials of the form \ $p(t) = \sum_{j=1}^{n}
a_{j}e^{i\lambda_{j} t}$\ where $n\in\NN,\;a_{j}\in{\cal B}$ and
$\lambda_{j}\in\RR$.
There is another way of viewing almost periodic functions which
is in keeping with the point of
view we have adopted in treating
quasiperiodic functions.
For any almost periodic function $\varphi\in{\cal AP}(\RR,
{\cal B})$
there is a curve $\Gamma :\RR\rightarrow \TT^{\infty}$,
dense in $\TT^{\infty}$ ($\TT^{\infty}$ is the compact abelian
group $\Pi_{j=1}^{\infty}S^{1}$), and
a continuous function $\gamma : \TT^{\infty}\rightarrow
{\cal B}$ (the {\em generating function} of $\varphi$) such
that $\varphi(t) = \big(\gamma\circ\Gamma
\big)(t)$. Then, \mbox{$\ergint \varphi\;dt\;=\;\int_{\TT^{\infty}}
\gamma\;d\mu$}\ where $d\mu$ is the (normalized)
Haar measure on $\TT^{\infty}$ (see [HR] or [DS]).
When almost periodic functions are looked at this way, from
the point of view of
generating functions, their special nature
becomes apparent. We are referring to their "compactness" in the
time variable:\ Although an almost periodic function is defined on
the entire real line,
the generating function is defined on the compact
space $\TT^{\infty}$,
in which the real line is embedded:
\ $\Gamma(\RR)\subset\TT^{\infty}$.
This is the
{\em Bohr compactification} of the real line
[HR].
Thus, almost periodic functions are "effectively" defined on the
compact space $\TT^{\infty}$
(through their generating function).
It is this fact which makes almost periodic functions a
generalization of periodic
functions, and is responsible for them sharing many
properties, an example of which is their uniform boundedness
as described below in Lemma 4.7.
This is a key property that we use to derive virial relations for
almost periodic solutions
(Theorem 4.10 ).
\setcounter{defn}{4}
\begin{defn}
A set $K\subset L^{p}(\RRn)$ is {\em uniformly bounded} (in
$L^{p}(\RRn)$) if,
for any $\ep >0$
there exists a ball $B_{\ep}\subset\RRn$ such that
\begin{equation}
\int_{B^{c}_{\ep}}\mid\varphi\mid^{p}\;\;<\;\:\ep
\;\;\;\;\;\forall\;\varphi\in K.
\end{equation}
\end{defn}
\begin{defn}
For a function $\varphi: \RR\rightarrow L^{p}(\RRn)$, the set
$\{\varphi(t)\,;\;t\in\RR\}$ is the {\em orbit} of
$\varphi$.
\end{defn}
\setcounter{lemma}{6}
\begin{lemma}
If $\varphi\in{\cal AP}(\RR; L^{p}(\RRn))$,
then the orbit of $\varphi$
is uniformly bounded in $L^{p}(\RRn)$.
\end{lemma}
\noindent\underline{Proof}:
A basic property of almost periodic functions is that their orbits
are relatively compact ([LZ] pp.2). Relatively compact subsets of
$L^{p}(\RRn)$ are uniformly bounded ([DS] Thm. IV.8.21)\ \ $\Box$
\vspace{.2in}
\noindent\underline{Remark:}\ \ That is, if $\varphi\in
{\cal AP}(\RR,L^{p}(\RRn))$, then for any $\ep>0$ there exists
a ball $B_{\ep}\subset\RRn$ such that
$$
\int_{B_{\ep}^{c}}|\varphi(x,t)|^{p}\;\;<\;\;\ep
$$
for all $t$.
\vspace{.2in}
In this section we are treating
the almost periodic function $\varphi$ as the primary
object rather than its
generating function $\gamma$.
Consequently, we have lost the ability to apply
Lebesgue's dominated convergence theorem which was the main tool
in the proof of Theorem 2.7. This is because $\varphi$ is defined only
on a set $\Gamma(\RRn)$ of measure zero in $\TT^{\infty}$ so that
pointwise convergence of $\varphi$ on $\Gamma(\RRn)$
(that is, pointwise
convergence in $t$) does not
imply the convergence of
the integral of its generating function over $\TT^{\infty}$.
For the proof of Theorem 4.10, below, in place of dominated
convergence we use the fact that the orbit of $\varphi$ is
uniformly bounded.
These remarks apply also to the quasiperiodic case. That is, we
could not use dominated convergence in Theorem 2.7 if we were treating
$\varphi$ instead of $\gamma$ there.
\setcounter{defn}{7}
\begin{defn} For $\varphi\in{\cal AP}(\RR, {\cal B})$
the mean value of $\varphi$ is
denoted by ${\cal M}(\varphi)\;\equiv \; \ergint \varphi(t)$.
\end{defn}
\noindent\underline{Remark}:\ \ Because
almost periodic functions are bounded (in ${\cal B}$),
${\cal M}(\varphi)$ is finite.
\setcounter{lemma}{8}
\begin{lemma}
Let $\varphi\in{\cal AP}_{q}$. Then
(i)\ \ $\varphi,\;\n\varphi\;\in\apll$,
(ii)\ \ $\hat{\varphi},\;\widehat{\n\varphi}\;\in\apll$,
where $\;\hat{ }\;$ denotes
the Fourier transform,
(iii)\ \ $\Rrep\varphi\;\in\aph$, where $\Rrep$ is as in Theorem 2.7,
(iv)\ \ $\|\partial_{t}\varphi\|_{2}^{m}\;,\|\n\varphi\|_{2}^{m}\;
\in \apr$ for $m\in\ZZ > 0$,
(v)\ \ if $h$ is a differentiable function on $\RRn$
that is bounded along with its
derivatives then \newline\indent \ \ \ \ \ \ $h\varphi\in\aph$,
(vi)\ \ if $f$ satisfies (H1) then $f(\varphi)\in\apll$ and
$F(\varphi)\in\apl$.
\end{lemma}
\noindent\underline{Proof}:
If $L: {\cal B}_{1}\rightarrow {\cal B}_{2}$ is
a continuous
map between Banach spaces and
$\varphi\in {\cal AP}(\RR,{\cal B}_{1})$, then
$L(\varphi)\in {\cal AP}(\RR, {\cal B}_{2})$ ([LZ] pp. 3).
Parts $(i) - (v)$ are a straightforward consequence of this.
Similarly for
part $(vi)$ after realizing that if $f$ satisfies (H1) then
the maps $f:\;L^{2}(\RRn)\cap L^{q}(\RRn)\rightarrow L^{2}(\RRn)$
and $F:\;L^{2}(\RRn)\cap L^{q}(\RRn)\rightarrow L^{1}(\RRn)$
are continuous (this follows from the proof of Theorem I.2.1 in
[K]).
\subsection{Virial relations for almost periodic solutions}
\setcounter{thm}{9}
\begin{thm}
Let $\varphi\in{\cal AP}_{q}$, for some $q\in [2,\infty]$,
satisfy (4.5),
$f$ satisfy (H1) with this same $q$, and $v$
be a smooth vector field on $\RRn$ satisfying the hypothesis of
Theorem 2.7. Then
\begin{equation}
\ergint\int_{\RR^{N}}
\left\{tr\,dv\; \big(\frac{1}{2}(\partial_{t}\varphi)^{2}
- F(\varphi)\big)\;+\;
\;\nabla\varphi\cdot[dv-\frac{1}{2}tr\,dv{\bf 1}]\,\nabla\varphi
\right\}\;\;=\;\;0.
\end{equation}
\end{thm}
\noindent\underline{Proof}:
Our approach begins as in Theorem 2.7. Set
\begin{equation}
\psi_{\ep,\delta}\equiv \vd\cdot\n\R2rep\varphi,
\end{equation}
where $\vd$ and $\Rrep$ are as in Theorem 2.7. Then $\psi_{\ep,\delta}
\in{\cal AP}\;\;\forall\ep,\delta> 0$ and so (cf. equation (4.5))
\begin{equation}
\ergint\int_{\RRn}
\Big\{-\partial_{t}\varphi\partial_{t}\psi_{\ep,\delta} +
\n\varphi\cdot\n\psi_{\ep,\delta} +
f(\varphi)\psi_{\ep,\delta}\Big\}\;\;=\;\;0\;\;\;\;\forall
\ep,\delta >0.
\end{equation}
We write
\begin{eqnarray}
\partial_{t}\varphi\partial_{t}\psi_{\ep,\delta} =
\partial_{t}\varphi\partial_{t}\vd\cdot\n\R2rep\varphi & = &
\partial_{t}\varphi\vd\cdot\n\R2rep\partial_{t}\varphi
\nonumber \\
& = &
\partial_{t}\varphi\Rrep\vd\cdot\n\Rrep\partial_{t}\varphi\;\;-
\;\;\partial_{t}\varphi[\Rrep,\vd]\cdot\n\Rrep\partial_{t}\varphi.
\end{eqnarray}
For the first term on the right,
\begin{eqnarray}
\int_{\RRn}
\partial_{t}\varphi\Rrep\vd\cdot\n\Rrep\partial_{t}\varphi & = &
\int_{\RRn}
\Rrep\partial_{t}\varphi\vd\cdot\n\Rrep\partial_{t}\varphi
\;\;\;\;\;\;\makebox{ (self-adjointness of $R_{\ep}$)}
\nonumber \\
& = &
\frac{1}{2}\int_{\RRn}\n\cdot(\vd(\Rrep\partial_{t}\varphi)^{2})
\;\;-\frac{1}{2}\int_{\RRn}\n\cdot\vd(\Rrep\partial_{t}\varphi)^{2}
\nonumber \\
& = &
-\frac{1}{2}\int_{\RRn}
\n\cdot v(\partial_{t}\varphi)^{2}
\;\;-\frac{1}{2}\int_{\RRn}
(\n\cdot\vd - \n\cdot v)(\partial_{t}\varphi)^{2}
\nonumber \\
& &\;\;\;\;\;\;-
\int_{\RRn}
\n\cdot\vd[(\Rrep-1)\partial_{t}\varphi]\partial_{t}\varphi
\nonumber \\
& & \;\;\;\;\;\;-\frac{1}{2}\int_{\RRn}
\n\cdot\vd[(\Rrep -1)\partial_{t}\varphi]^{2}.
\end{eqnarray}
Let $B(\rho)\subset\RRn$ denote the ball of radius $\rho$.
By Lemma 4.7, for any $\eta > 0$ there exists a $\rho > 0$ such that
$\int_{B^{c}(\rho)}(\partial_{t}\varphi)^{2}\;<\;\eta/\|
\n\cdot\vd - \n \cdot v\|_{\infty}$ for all $t$.
Because $\n\cdot\vd\rightarrow \n\cdot v$ uniformly on
compact sets there is a $\delta_{o}(\eta) > 0$ such that
$\mid\!\n\cdot\vd - \n\cdot v\!\mid\;<\,\eta$
on $B(\rho)$ for $\delta < \delta_{o}$.
Then, for any such $\delta$ and any $T$,
\begin{eqnarray}
\Big|\frac{1}{T}\int_{0}^{T}\int_{\RRn}
(\n\cdot\vd - \n\cdot v)(\partial_{t}\varphi)^{2}\Big|&\leq&
\frac{1}{T}\int_{0}^{T}\int_{B(\rho)}
\mid\n\cdot\vd - \n\cdot v\mid(\partial_{t}\varphi)^{2}
\nonumber \\
&\;\;\;\;\;\;\;\;\;\;+&
\frac{1}{T}\int_{0}^{T}\int_{B^{c}(\rho)}
\mid\n\cdot\vd - \n\cdot v\mid(\partial_{t}\varphi)^{2}
\nonumber \\
&\leq&
\eta\frac{1}{T}\int_{0}^{T}\int_{\RRn}
(\partial_{t}\varphi)^{2}\;\;\;\;\;\;+\;\;\;\eta.
\end{eqnarray}
Taking the limit $T\rightarrow\infty$,
\begin{equation}
\Big|\ergint\int_{\RRn}
(\n\cdot\vd - \n\cdot v)(\partial_{t}\varphi)^{2}\Big|\;\;\leq
\;\;\eta{\cal M}(\;\|\partial_{t}\varphi\|_{2}^{2}) + \eta.
\end{equation}
Since $\eta$ was arbitrary we conclude that
\begin{equation}
\lim_{\delta\rightarrow 0}\;\;\ergint\int_{\RRn}
(\n\cdot\vd - \n\cdot v)(\partial_{t}\varphi)^{2} = 0.
\end{equation}
Bounding $\|\n\cdot\vd\|_{\infty}\leq c$ uniformly in $\delta$,
\begin{equation}
\Big|\ergint\int_{\RRn}
\n\cdot\vd [(\Rrep -1)\partial_{t}\varphi]\partial_{t}\varphi\Big|
\;\;\leq \;\;\ergint c\|(\Rrep-1)\partial_{t}\varphi\|_{2}
\|\partial_{t}\varphi\|_{2}.
\end{equation}
Using the Fourier transform,
\begin{equation}
\|(\Rrep-1)\partial_{t}\varphi\|_{2}^{2} =
\int_{\RRn}\frac{(\ep^{2}p^{2})^{2}}{(1 + \ep^{2}p^{2})^{2}}
(\partial_{t}\hat{\varphi}(p))^{2}
=\int_{\RRn}h_{\ep}(p)(\partial_{t}\hat{\varphi}(p))^{2}.
\end{equation}
Since $h_{\ep}\rightarrow 0$ as $\ep\rightarrow 0$
uniformly on compact sets
with $h_{\ep}(p)\leq 1$, and since
$\{\partial_{t}\hat{\varphi};\;t\in\RR\}$ is uniformly
bounded in $L^{2}(\RRn)$,
applying the same argument as above we conclude
that, for any $\eta^{2}> 0$ and $\ep $ sufficiently small,
\begin{eqnarray}
\|(\Rrep -1)\partial_{t}\varphi\|_{2}^{2} &\leq&
\eta^{2}\|\partial_{t}\varphi\|^{2}_{2} + \eta^{2} \nonumber \\
&\leq&
\eta^{2}(\|\partial_{t}\varphi\|_{2} + 1)^{2}\;\;\;\;\forall t.
\end{eqnarray}
Therefore,
\begin{eqnarray}
\ergint\|(\Rrep -1)\partial_{t}\varphi\|_{2}\|\partial_{t}\varphi\|
_{2} &\leq&
\ergint\eta(\|\partial_{t}\varphi\|_{2} + 1)\|\partial_{t}\varphi\|
_{2} \nonumber \\
&=&
\eta\left({\cal M}(\|\partial_{t}\varphi\|^{2}_{2})+
{\cal M}(\|\partial_{t}\varphi\|_{2})\right),
\end{eqnarray}
and so
\begin{equation}
\lim_{\ep\rightarrow 0}\;\;\;\;\ergint\int_{\RRn}
\n\cdot\vd[(\Rrep-1)\partial_{t}\varphi]\partial_{t}\varphi\; = \;0
\end{equation}
uniformly in $\delta$.
The last term in equation (4.11) is treated in a similar way while
the second term on the right hand side of equation (4.10) can be treated
as below (see equation (4.25));\ the result being that both of these
terms converge to zero.
Thus,
\begin{equation}
\lim_{\ep,\delta\rightarrow 0}\;\;\;\;\ergint\int_{\RRn}
\partial_{t}\varphi\partial_{t}\psi_{\ep,\delta}\;\;=
\;\;\;\ergint\int_{\RRn}-\frac{1}{2}tr\,dv\,(\partial_{t}\varphi)^{2}.
\end{equation}
Considering the next term in equation (4.9),
\begin{eqnarray}
\n\varphi\cdot\n\psi_{\ep,\delta} &=& \n\varphi\n\vd\cdot\n\R2rep\varphi
\nonumber \\
&=&
\n\varphi\vd\cdot\n{\bf 1}\,\n\R2rep\varphi + \n\varphi\,d\vd\,
\n\R2rep\varphi.
\end{eqnarray}
We write the first term on the right hand side as
\begin{equation}
\n\varphi\vd\cdot\n{\bf 1}\,\n\R2rep\varphi =
\n\varphi\Rrep\vd\cdot\n{\bf 1}\,\n\Rrep\varphi -
\n\varphi[\Rrep,\vd]\cdot\n{\bf 1}\,\Rrep\n\varphi.
\end{equation}
Furthermore,
\begin{eqnarray}
\n\varphi\Rrep\vd\cdot\n{\bf 1}\,\n\Rrep\varphi &=&
-\frac{1}{2}\n\cdot\vd\mid\n\Rrep\varphi\mid^{2}
+ \frac{1}{2}\n\cdot(v_{\delta}|\n R_{\ep}\varphi|^{2})
\nonumber \\
&=&
-\frac{1}{2}\n\cdot v\mid\n\varphi\mid^{2} -
\frac{1}{2}[\n\cdot\vd - \n \cdot v]\mid\n\varphi\mid^{2}
\nonumber \\
&\;\;\;\;\;-&
\n\cdot\vd[(\Rrep-1)\n\varphi]\n\varphi -
\frac{1}{2}\n\cdot\vd[(\Rrep-1)\n\varphi]^{2}
\nonumber \\
&\;\;\;\;\;+&
\frac{1}{2}\n\cdot(v_{\delta}|\n R_{\ep}\varphi|^{2}).
\end{eqnarray}
Ignoring the divergence term, the terms on the
right can be treated in the
same way as was done for equation (4.11) whence we conclude that
\begin{equation}
\lim_{\ep,\delta\rightarrow 0}\;\;\ergint\int_{\RRn}
\n\varphi\Rrep\vd\cdot\n{\bf 1}\,\n\Rrep\varphi =
\ergint\int_{\RRn}
-\frac{1}{2}tr\,dv\,\mid\n\varphi\mid^{2}.
\end{equation}
Considering now
the second term on the right hand side of equation (4.22), we
write
\begin{equation}
\int_{\RRn}
\n\varphi[\Rrep,\vd]\cdot\n{\bf 1}\,\Rrep\n\varphi =
\int_{\RRn}
\n\varphi\Rrep\ep^{2}(
\Delta\vd\cdot\Rrep\n{\bf 1} +
2\n\vd\n\cdot\Rrep\n{\bf 1})\Rrep\n\varphi
\end{equation}
(cf. equation (2.15)).
We have
\begin{equation}
\Big|\int_{\RRn}
\ep^{2}\n\varphi\Rrep\Delta\vd\cdot\Rrep\n{\bf 1}\Rrep\n\varphi\Big|
\;\;\leq\;\;c\ep\|\n\varphi\|_{2}^{2}
\end{equation}
(cf. equation (2.18))
so that this term will converge to zero in the time mean as
$\ep\rightarrow 0$, uniformly in $\delta$.
The other term in equation (4.25) we treat in a similar way as in
equation (2.19);
\begin{eqnarray}
\Big|\int_{\RRn}
\ep^{2}\n\varphi\Rrep\n\vd\n\cdot\Rrep\n{\bf 1}\,\Rrep\n\varphi\Big|
&\leq& c\|\n\varphi\|_{2}\|\ep^{2}\n^{2}{\bf 1}\,\Rrep\n\varphi\|
_{2} \nonumber \\
&\leq& c\|\n\varphi\|_{\infty,2}
\|\ep^{2}\n^{2}{\bf 1}\,\Rrep\n\varphi\|_{2}
\end{eqnarray}
where $c$ is independent of $\delta$.
Note that $\|\n\varphi\|_{\infty,2}$ is a constant.
Passing to the Fourier transform, for $|a| = 2$
and $\psi\in\apll$,
\begin{equation}
\|\ep^{2}\partial^{a}\Rrep\psi\|^{2}_{2} =
\int_{\RRn}
\frac{\ep^{4}\mid p^{a}\mid^{2}}{(1 + \ep^{2}p^{2})^{2}}
\mid\hat{\psi}(p,t)\mid^{2}.
\end{equation}
The function $\ep^{4}\mid\!p^{a}\!\mid^{2}
(1 +\ep^{2}p^{2})^{-2}$ converges to zero uniformly on compact
sets and is bounded by a constant $c^{'}$ that depends only on $a$
while $\hat{\psi}(t)$
is uniformly bounded (in $L^{2}(\RRn)$)
so that, by the same arguments as made before (cf. equation (4.17)),
for
any $\eta>0$ and sufficiently small ~$\ep$,
\begin{equation}
\|\ep^{2}\partial^{a}\Rrep\psi\|_{2}^{2}\;\;\leq
\;\;\eta^{2}(\|\psi\|_{2} + c^{'})^{2}\;\;\;\;\forall\; t.
\end{equation}
Thus,
\begin{equation}
\ergint c\|\n\varphi\|_{\infty,2}\|\ep^{2}\n^{2}{\bf 1}\,\Rrep
\n\varphi\|_{2}\;\;\leq
\;\;\eta c\|\n\varphi\|_{\infty,2}{\cal M}(\|\n\varphi\|^{2}_{2} +
c^{'})
\end{equation}
and so
\begin{equation}
\lim_{\ep,\delta\rightarrow 0}\;\;\;\ergint\int_{\RRn}
\ep^{2}\n\varphi\Rrep\n\vd\n\cdot\Rrep\n{\bf 1}\,\Rrep\n\varphi\;\;=
\;\;0.
\end{equation}
The second term in equation (4.21) we write as
\begin{equation}
\n\varphi d\vd\n\R2rep\varphi = \n\varphi\,dv\,\n\varphi +
\n\varphi(d\vd - dv)\n\varphi + \n\varphi\,d\vd\,(\R2rep -1)\n\varphi.
\end{equation}
Applying Schwarz's inequality to the second term on the right
we find that
\begin{equation}
\|\n\varphi(d\vd - dv)\n\varphi\|_{1} \leq
\|\n\varphi\|_{2}\|(d\vd - dv)\n\varphi\|_{2}.
\end{equation}
Since $(d\vd - dv)\;\rightarrow 0$ uniformly on compact sets with
$\|d\vd -dv\|_{\infty}$ uniformly bounded, by familiar arguments
we conclude that for any $\eta>0$ and
$\delta$ sufficiently small,
\begin{equation}
\|(d\vd - dv)\n\varphi\|_{2} \leq \eta(\|\n\varphi\|_{2} + c)
\end{equation}
so that
\begin{equation}
\ergint \|\n\varphi(d\vd -dv)\n\varphi\|_{1} \;\;\leq
\;\;\eta\left({\cal M}(\|\n\varphi\|_{2}^{2} +
c{\cal M}(\|\n\varphi\|_{2})\right)
\end{equation}
and hence
\begin{equation}
\lim_{\delta\rightarrow 0}\;\;\ergint\int_{\RRn}
\n\varphi(d\vd - dv)\n\varphi\;\;=\;\;0.
\end{equation}
The third term on the right hand side of equation (4.32) is treated
analogously by noting that $\|d\vd\|_{\infty}$ is uniformly
bounded and then using the Fourier transform to show that
\newline$\|(\R2rep -1)\n\varphi\|_{2}\leq\eta(\|\n\varphi\|_{2} + 1)
\forall t$
for any $\eta>0$ and $\ep$ sufficiently small (cf. (4.17)).
Thus,
\begin{equation}
\lim_{\ep,\delta\rightarrow 0}\;\;\;\ergint\int_{\RRn}
\n\varphi\,dv\,\n\R2rep\n\varphi\;\;=\;\;\ergint
\int_{\RRn}\n\varphi\,dv\,\n\varphi.
\end{equation}
To treat the nonlinear term we begin by writing
\begin{eqnarray}
\int_{\RRn} f(\varphi)\psi_{\ep,\delta} & = &
-\int_{\RRn}\n\cdot v\,F(\varphi) +
\int_{\RRn}(\n\cdot\vd - \n\cdot v)F(\varphi) \nonumber \\
&\;\;\;\;\;\;\;\;+&\int_{\RRn}
f(\varphi)\vd\cdot(\R2rep - 1)\n\varphi
\end{eqnarray}
Since $F(\varphi)\in\apl$ (Lemma 4.9) with $(\n\cdot\vd - \n v)
\rightarrow
0$ uniformly on compact sets we conclude, by
arguments made before, that
\begin{equation}
\lim_{\delta\rightarrow 0}\;\;\ergint\int_{\RRn}
(\n\cdot\vd - \n v)F(\varphi)\;\;=\;\;0.
\end{equation}
Using the Fourier transform as before we can show that
\begin{equation}
\|(\R2rep -1)\n\varphi\|_{2}\;\;\leq\;\;\eta
(\|\n\varphi\|_{2} + 1)\;\;\;\;\forall\;t
\end{equation}
for any $\eta>0$ and $\ep$ sufficiently small. Now apply
Schwarz's inequality to the third term on the right hand side of
equation (4.38) to conclude that
\begin{equation}
\Big|\int_{\RRn}f(\varphi)\vd\cdot(\R2rep -1)\n\varphi\Big|\;\;\leq
\;\;\eta\|\vd\|_{\infty}\|f(\varphi)\|_{2}(\|\n\varphi\|_{2} + 1).
\end{equation}
Both $\|f(\varphi)\|_{2}$ and $(\|\n\varphi\|_{2} + 1)$ are numerical
almost periodic functions (Lemma 4.9), i.e., are members of
${\cal AP}(\RR,\RR)$ and hence their product is
almost periodic ([LZ] pp 6). Then, for any $\eta>0$ and $\ep$
sufficiently small,
\begin{equation}
\Big|\ergint\int_{\RRn}f(\varphi)\vd\cdot(\R2rep -1)\n\varphi
\Big|\;\;\leq
\;\;\eta\|\vd\|_{\infty}{\cal M}\Big(\|f(\varphi)\|_{2}
(\|\n\varphi\|_{2}
+ 1)\Big).
\end{equation}
Since $\|v_{\delta}\|_{\infty}{\cal M}\Big(\|f(\varphi)\|_{2}
(\|\n\varphi\|_{2} + 1)\Big)$ is bounded uniformly in $\delta$,
we have that
\begin{equation}
\lim_{\ep\rightarrow 0}\;\;\ergint\int_{\RRn}
f(\varphi)\vd\cdot(\R2rep - 1)\n\varphi\;\;=\;\;0
\end{equation}
uniformly in $\delta$.
Thus,
\begin{equation}
\ergint\int_{\RRn}
f(\varphi)\psi_{\ep,\delta}\;\;=
\;\;\ergint\int_{\RRn}-tr\,dv\,F(\varphi)\;\;\;\;\;\;+\;\;\;E(\ep,
\delta)
\end{equation}
where $E(\ep,\delta)$ can be made arbitrarily small first by choosing
$\delta$ and then $\ep$ sufficiently small.
This, along with equations (4.9), (4.20), (4.24) and (4.37), completes
the proof of Theorem 4.10\ \ $\Box$
\section{Nonexistence of almost periodic solutions}
\setcounter{thm}{0}
\setcounter{equation}{0}
We use the virial relation
(4.7) to prove a nonexistence theorem for almost periodic
solutions. A special case of this result
($N=1$ and $c=\frac{1}{2}$)
was proven in [SV].
\begin{thm}
(Nonexistence of almost periodic solutions of NLW)\ \ Let
$\varphi\in {\cal AP}_{q}$ for some $q\in [2,\infty]$ be a
weak almost periodic
solution of NLW with $f$
satisfying (H1) with this same $q$ and, for
some $c\in[\frac{N-2}{2N},\frac{1}{2}]$,
the inequality
$$
F(z) - czf(z)\leq 0\;\;\mbox{ for all } z\mbox{ such that }
\mid z\mid\leq\|\varphi\|_{L^{\infty}
(\RR^{N+1})}.
$$
Then $\varphi$ is independent of time.
\end{thm}
\noindent\underline{Proof}:
As in Proposition 3.3,
we use the vector field associated to dilations on $\RRn$ along with
the Gauge transformation to
derive the following identity valid
for any $c\in\RR$ (we use Lemma~4.9, part $(v)$ for the Gauge
transformation);
\begin{equation}
\ergint\int_{\RRn}
\left\{(\frac{1}{2}-c)(\partial_{t}\varphi)^{2}
+ (c + \frac{2-N}{2N})\mid\n\varphi\mid^{2}\right\} =
\ergint\int_{\RRn}
\{F(\varphi) - c\varphi f(\varphi)\}.
\end{equation}
By hypothesis
both sides of this equation must be zero. In the case
$c\in(\frac{N-2}{2N},\frac{1}{2}]$ this implies that
\begin{equation}
\ergint\int_{\RRn}\mid\n\varphi\mid^{2} \;\;= \;\;0 \;\;= \;\;\ergint
\|\n\varphi\|^{2}_{2}.
\end{equation}
Parseval's relation for Hilbert space valued almost periodic
functions ([LZ] pp 31) states that
\begin{equation}
\ergint\|\n\varphi\|_{2}^{2} = \sum_{k=1}^{\infty}
\|a_{k}\|_{2}^{2}
\end{equation}
where $\sum_{k=1}^{\infty}a_{k}e^{i\omega_{k}t}$ is the Fourier
series associated to $\n\varphi$.
By the uniqueness of these
series it follows from equations (5.2) and (5.3) that
$\n\varphi(t) = 0$
(in $L^{2}(\RRn)$) for all $t$ which implies that,
since $\varphi(t)\in
L^{2}(\RRn)$, \ $\varphi(t)= 0$ for all $t$.
If $c=\frac{N-2}{2N}$ then a similar argument applied to
$\partial_{t}\varphi$ leads to the
conclusion that $\partial_{t}\varphi(t)
= 0$ (in $L^{2}(\RRn)$) for all $t$. That is, $\varphi(t) =
\psi$ for some $\psi\in L^{2}(\RRn)$ and for all $t$\ \ $\Box$
\vspace{.2in}
\noindent\underline{Remark}:\ Example 1 in Section 3 and
Corollary 3.6 hold also for
almost periodic solutions, and by the same proofs as presented there.
\newpage
\setcounter{lemma}{0}
\setcounter{equation}{0}
\setcounter{defn}{0}
\renewcommand{\theequation}{\Alph{section}.\arabic{equation}}
\renewcommand{\thedefn}{\Alph{section}.\arabic{defn}}
\renewcommand{\thelemma}{\Alph{section}.\arabic{lemma}}
\section{Appendix}
\setcounter{section}{1}
In this appendix we illustrate how virial relations for NLW
can be derived and understood in a natural way
by formulating NLW as a variational
problem. From this aspect the vector fields
characterizing the virial relations derived in this paper
define infinitesimal generators of transformation groups on
the space $H^{1}(\RRn)$ on which the action functional is defined.
Our purpose here is to describe virial relations and conservation
laws from a common point of view;\ in terms of
the behavior of the action
under such transformation groups.
All transformation groups generate virial relations. If the
transformation group happens to leave the action invariant, i.e.,
is a symmetry group, then
the associated virial relation is "trivial" in the sense that
it is merely the consequence of the fact that the integrand is
a divergence. At the same time from this one can infer
a conservation law for the associated Euler-Lagrange
equation (Noether's Theorem).
For general transformation groups the integrand is not necessarily
a divergence. How far from a divergence it is produces
the "nontrivial" virial relations derived in this paper.
\vspace{.3in}
\noindent\underline{Variational Calculus and Transformation
Groups}
\vspace{.2in}
Let $X$ be a Banach space of functions defined on $\RR^{m}$ and $S$ a
functional $X\rightarrow \RR$ of the form
\begin{equation}
S[\varphi] = \int_{\Omega}{\cal S}(\varphi,\nabla_{m}\varphi)
\end{equation}
where
${\cal S}\,:\,\RR\times\RR^{m}\rightarrow\RR$,\ $(u,p)
\rightarrow
{\cal S}(u,p)$ is the Lagrangian,
\ $\nabla_{m} $ denotes the gradient with respect to all
the variables, and $\Omega\subset\RR^{m}$. We consider $S$ as
being associated to a differential equation
$K(\varphi)=0$ through the relation
$S'[\varphi]=0\Leftrightarrow\varphi$ is a (weak) solution of
$K(\varphi) = 0$, where
$S'[\varphi]$ denotes the Fr\'{e}chet derivative.
In this case $K(\varphi) = 0$ is the Euler-Lagrange equation
associated to $S$;
\begin{equation}
K(\varphi) =
{\cal S}_{u} - \n_{m}\cdot{\cal S}_{p}\;=\;
\;\frac{\partial{\cal S}}
{\partial\varphi}-\frac{\partial}{\partial x_{i}}
\frac{\partial{\cal S}}{\partial\varphi_{i}} = 0,
\end{equation}
where $\varphi_{i}\equiv\frac{\partial\varphi}{\partial x_{i}}$ and
summation over $i = 1,\ldots,m$ is implied.
For example,
if we are interested in $l$-quasiperiodic solutions of NLW with
frequency $\omega$, then
we formulate a variational
problem on a space of (generating) functions defined
on $\Omeg$ by defining
the Lagrangian,
\begin{equation}
{\cal S} = {\cal S}(\gamma,\n\gamma,\D\gamma) =
-\frac{1}{2}(\Dw\gamma)^{2}
+ \frac{1}{2}\mid\nabla\gamma\mid^{2}
+ F(\gamma).
\end{equation}
>From this formula we see that a natural choice
for $X$ is $H^{1}(\Omeg)$. Then,
\begin{equation}
S[\gamma]\;=\;\int_{\Omeg}\Big\{
-\frac{1}{2}(\Dw\gamma)^{2} + \frac{1}{2}|\n\gamma|^{2} + F(\gamma)
\Big\}
\end{equation}
and
\begin{equation}
S'[\gamma](\beta) = \int_{\Omeg}
\Big\{-\Dw\gamma\Dw\beta
+ \nabla\gamma\cdot\nabla\beta + f(\gamma)\beta\Big\}.
\end{equation}
(cf. equation (2.2)).
To formulate NLW as a variational problem for almost periodic
solutions $\varphi\in{\cal AP}$
we define the action,
\begin{equation}
S[\varphi]\;\;=\;\;\ergint\int_{\RRn}
\Big\{-\frac{1}{2}(\partial_{t}\varphi)^{2} +
\frac{1}{2}\mid\n\varphi\mid^{2} + F(\varphi)\Big\}
\end{equation}
so that
\begin{equation}
S'[\varphi](\psi) = \ergint\int_{\RRn}
\Big\{-\partial_{t}\varphi\partial_{t}\psi +
\n\varphi\cdot\n\psi - f(\varphi)\psi\Big\}
\end{equation}
(cf. equation (4.5) ).
We can show that the actions (A.4) and (A.6)
are positive at critical
points;
\begin{equation}
S'[\gamma]\;=\;0\;\Longrightarrow\;S[\gamma]\;>\;0
\;\;\mbox{ (cf. equation (A.4))}
\end{equation}
and
\begin{equation}
S'[\varphi]\;=\;0\;\Longrightarrow\;S[\varphi]\;>\;0
\;\;\mbox{ (cf. equation (A.6)) }.
\end{equation}
This follows from the virial relation associated to the dilations
in $x$ (equations (3.4) and (5.1) with $c=0$) from which we
infer that
\begin{equation}
\int_{\Omeg}F(\gamma)\;=\;\int_{\Omeg}\Big\{
\frac{1}{2}(\Dw \gamma)^{2}
+ \frac{2-N}{2N}|\n\gamma|^{2}\Big\}
\end{equation}
and
\begin{equation}
\ergint\int_{\RRn} F(\varphi)\;=\;\ergint\int_{\RRn}
\Big\{
\frac{1}{2}(\partial_{t}\varphi)^{2} +
\frac{2-N}{2N}|\n\varphi|^{2}\Big\},
\end{equation}
for solutions $\gamma$ of NLW on $\Omeg$ and almost periodic
solutions $\varphi$ of NLW on $\RR^{N+1}$ respectively.
Substituting these into (A.4) and (A.6) justifies the claims
(A.8) and (A.9).
\vspace{.2in}
Returning to the abstract set-up,
variations of $S$ are defined through transformation groups
acting on $X$.
Let $T_{\lambda}:\,X\rightarrow X ;\;\varphi\mapsto
\varphi_{\lambda}\equiv T_{\lambda}\varphi$, with
$T_{0}={\bf 1}$, be a
strongly continuous 1-parameter group of
transformations with infinitesimal generator $A$,\ $A:\, D(A)\subset
X \rightarrow X$, $D(A)$ denoting the
domain of $A$. Here $A$ is defined by
\ $A\varphi = \frac{d}{d\lambda}\varphi_{\lambda}
\Big|_{\lambda = 0}$.
If $\varphi\in D(A)$ is a
critical point of $S$,
then applying the chain rule to the function
$S[\varphi_{\lambda}]:\,\RR\rightarrow\RR$ we find that
\begin{equation}
\frac{d}{d\lambda}S[\varphi_{\lambda}]\Big|_{\lambda=0}\;\;=\;
\;S'[\varphi](A\varphi)\;=\;0.
\end{equation}
If $A$ is a differential operator then
this equation is an integral formula involving the
solution $\varphi$ and its derivatives.
It is the virial relation associated
to the transformation $T_{\lambda}$
and corresponds to equation (2.4) (Theorem ~2.7), as we will
describe in more detail below.
A particular class of transformations on $X$ arise from
diffeomorphisms of $\RR^{m}$. Suppose $v:
\RR^{m}\rightarrow\RR^{m}$ is a vector field on $\RR^{m}$ that
generates a global flow $\Phi_{\lambda}:\RR\times\RR^{m}\rightarrow
\RR^{m}$,
($\Phi_{0} = {\bf 1}$).
Then the map
$\varphi\mapsto\varphi_{\lambda}
\equiv\varphi\circ\Phi_{\lambda}\equiv
T_{\lambda}\varphi$
defines a 1-parameter group of transformations $T_{\lambda}$ on $X$
with infinitesimal generator
$v\cdot\nabla_{m}$. For the remainder of the
appendix we will always assume that $T_{\lambda}$
is of this form.
Formally,
\begin{equation}
\frac{d}{d\lambda}S[\varphi_{\lambda}]\Big|_{\lambda=0}\;\;=
\;S'[\varphi]
(v\cdot\nabla_{m}\varphi)\;=\;\int_{\Omega}
\Big\{ {\cal S}_{u}(v\cdot\nabla_{m}\varphi)\;+
\;{\cal S}_{p}\cdot\nabla_{m}(v\cdot\n_{m}\varphi)
\Big\}.
\end{equation}
We use the word formally here because $v\cdot\n_{m}\varphi$
may not lie in $X$
(that is, $\varphi$ may not lie in $D(v\cdot\n_{m})$) in which
case a regularization procedure is required, as was done in
the proof of Theorem 2.7.
If $\varphi$ is a critical point of $S$,
then we have the virial relation
\begin{equation}
0\;=\;\int_{\Omega}
\Big\{ {\cal S}_{u}(v\cdot\nabla_{m}\varphi)\;+
\;{\cal S}_{p}\cdot\nabla_{m}(v\cdot\n_{m}\varphi)\Big\}.
\end{equation}
We will see that this equation is precisely the virial relation
(2.4) of Theorem 2.7 corresponding to the vector field $v$.
The hypothesis of Theorem 2.7 concerning vector fields on
$\RRn$ (cf. (2.3)) guarantees that these vector fields generate
global flows on $\RRn$, which in turn defines transformation
groups on $H^{1}(\Omeg)$.
We now consider how such transformation groups affect the action
$S$ and in this way we will distinguish between transformation
groups that preserve $S$ or not and the consequences thereof for
the associated virial relations.
\vspace{.2in}
\noindent\underline{Symmetries, Conservation Laws, and Virial
Relations}
\vspace{.1in}
\begin{defn} $T_{\lambda}:X\rightarrow X$
is a {\em symmetry group of $S$} if
$S[\varphi_{\lambda}]=S[\varphi]$ for all $\varphi\in X$ and for
all $\lambda\in\RR$, where $\varphi_{\lambda}
\equiv T_{\lambda}\varphi = \varphi\circ\Phi_{\lambda}$.
\end{defn}
\setcounter{lemma}{1}
\begin{lemma}{\bf (Noether)}\ \ If
$T_{\lambda}$ is a symmetry group
of $S$ with infinitesimal generator
$v\cdot\n_{m}$, then for any $\varphi\in X$ the expression
\begin{displaymath}
{\cal S}_{u}(v\cdot\nabla_{m}\varphi)\; +
\;{\cal S}_{p}\cdot\nabla_{m}
(v\cdot\nabla_{m}\varphi)
\end{displaymath}
is a divergence (cf. (A.13)).
\end{lemma}
In fact ([GF] Thm 2 \S 37),
\begin{equation}
{\cal S}_{u}(v\cdot\nabla_{m}\varphi)\; +
\;{\cal S}_{p}\cdot\nabla_{m}
(v\cdot\nabla_{m}\varphi)
\;=\;\n_{m}\cdot{\cal S}v.
\end{equation}
Therefore, if $T_{\lambda}$ is a symmetry group
of $S$ and $\varphi$
is a critical point of $S$,
then using the Euler-Lagrange equation (A.2) we obtain
from (A.15)
\begin{eqnarray}
0 &=& \;{\cal S}_{u}
(v\cdot\nabla_{m}\varphi)\; + \;{\cal S}_{p}\cdot\nabla_{m}
(v\cdot\n_{m}\varphi)\;
- \;\n_{m}\cdot{\cal S}v \nonumber \\
&=& \;({\cal S}_{u} - \n_{m}\cdot{\cal S}_{p})
v\cdot\nabla_{m}\varphi\; +
\;\nabla_{m}\cdot({\cal S}_{p}
v\cdot\nabla_{m}\varphi - {\cal S}v) \nonumber \\
&=&
\;\n_{m}\cdot ({\cal S}_{p}v
\cdot\n_{m}
\varphi - {\cal S}v).
\end{eqnarray}
We describe how this formula leads to a conservation law.
Consider NLW on $\Omeg$, where the action $S$ is defined
by (A.4). Recall that
$\n_{m} = (\n,\,\D)$. Writing $v = (v_{a},\,v_{b})$
where $v_{a}$ is a vector field on $\RRn$ and $v_{b}$ is a vector
field on $\TTl$,
the last line in (A.16) reads
\begin{equation}
0\;=\;\n\cdot\Big(
\frac{\partial{\cal S}}{\partial\n\gamma}
v\cdot\n_{m}\gamma-{\cal S}v_{a}\Big)\;+
\;\D\cdot\Big(
\frac{\partial{\cal S}}{\partial\D\gamma}v\cdot\n_{m}\gamma
-{\cal S}v_{b}\Big).
\end{equation}
Here $\partial{\cal S}/\partial\n\gamma$ and
$\partial{\cal S}/\partial\D\gamma$ denote the vectors
$(\partial{\cal S}/\partial\gamma_{1},\ldots,\partial{\cal S}/
\partial\gamma_{N})$ and
$(\partial{\cal S}/\partial\gamma_{N+1},\ldots,\partial{\cal S}/
\partial\gamma_{N+l})$ respectively, where
$\gamma_{i} = \partial\gamma/\partial x_{i}$, \
$i = 1,\ldots,N$, and $\gamma_{N+j} = \partial\gamma/
\partial \theta_{j}$,\ $j = 1,\ldots,l$.
If we set
\begin{equation}
{\bf E}_{v}(\gamma)\;\equiv\;\int_{\RRn}
\Big(\frac{\partial{\cal S}}{\partial\D\gamma}v\cdot\n_{m}\gamma
- {\cal S}v_{b}\Big),
\end{equation}
and if $(\partial{\cal S}/\partial\n\gamma)v\cdot\n_{m}\gamma -
{\cal S}v_{a}$
vanishes sufficiently rapidly as $|x|\rightarrow\infty$,
then from (A.17) and the divergence theorem we then have the
conservation law
\begin{equation}
{\cal D}\cdot{\bf E}_{v}(\gamma) = 0.
\end{equation}
As an example,
let $v_{T}$ be the infinitesimal generator of translation
on $\TTl$ along
$\Gamma_{\omega}$ (cf. Definition 2.1);
$v_{T} = (0,\,\omega)$. Then,
\begin{equation}
{\bf E}_{v_{T}}(\gamma) \;=\;-\omega\int_{\RRn}
\Big\{\frac{1}{2}(\Dw\gamma)^{2} + \frac{1}{2}|\n\gamma|^{2}
+ F(\gamma)\Big\}\;\equiv\;-\omega E(\gamma)
\end{equation}
where $E(\gamma)$ is the energy of $\gamma$.
In this case $\Dw E(\gamma) = {\cal D}\cdot{\bf E}_{v_{T}}(\gamma)
= 0$.
Therefore, $E(\gamma)$
is constant along $\Gamma_{\omega}$ ($\Dw$ is the directional
derivative along $\Gamma_{\omega}$).
If $E(\gamma)$ is continuous, then because $\Gamma_{\omega}$
is dense in
$\TTl$, \ $E(\gamma)$ is constant on $\TTl$.
Considering quasiperiodic solutions $\varphi$ of NLW;
\ $\varphi(x,t) = \gamma(x,\omega t)$, the argument just
given shows that the energy $E(\varphi)$
of $\varphi$ is conserved;
\begin{equation}
\frac{d}{dt}E(\varphi)\;=\;0\;\;\makebox{ where }
\;\;E(\varphi)\;\equiv
\;\int_{\RRn}\Big\{\frac{1}{2}(\partial_{t}\varphi)^{2}
+ \frac{1}{2}|\n\varphi|^{2} + F(\varphi)\Big\}.
\end{equation}
The energy of a solution $\gamma$, like
any functional of $\gamma$ that is constant
on $\TTl$,
can be used to derive an integral identity simply
by integrating it over $\TTl$;
\begin{equation}
(2\pi)^{l}E(\gamma)\;=\;\int_{\TTl}E(\gamma)\;
=\;\int_{\Omeg}\Big\{
\frac{1}{2}(\Dw\gamma)^{2} + \frac{1}{2}
|\n\gamma|^{2} + F(\gamma)\Big\}.
\end{equation}
Using (A.10) we derive the inequality
\begin{equation}
E(\gamma)\;=\;(2\pi)^{-l}\int_{\Omeg}
\Big\{(\Dw\gamma)^{2} + \frac{1}{N}|\n\gamma|^{2}\Big\}
\;\geq\;0,
\end{equation}
which shows that the energy of a quasiperiodic solution
is positive, and zero if and only if the solution is zero.
By performing the same arguments with the action defined by
(A.6) we can see that the energy of almost periodic solutions is
positive also.
\vspace{.2in}
We return to general transformations on $X$
(i.e., not necessarily symmetry groups of $S$).
For an arbitrary transformation group $T_{\lambda}$ the formula
(A.15) may not be true.
In this regard, to each $T_{\lambda}$ we define a function
$g$ on $X$,\ $g = g(\varphi,\n_{m}\varphi,v)$
where $v\cdot\n$ is the
infintesimal generator of $T_{\lambda}$, by the following
formula;
\begin{equation}
g\;\equiv
\;{\cal S}_{u}(v\cdot\nabla_{m}\varphi)\; +
\;{\cal S}_{p}\cdot\nabla_{m}
(v\cdot\nabla_{m}\varphi) - \n_{m}\cdot{\cal S}v.
\end{equation}
Then, when $T_{\lambda}$ is a symmetry group of $S$,
\ $g \equiv 0$ by Lemma A.2. Otherwise $g$ may not be zero.
This motivates us to think of the function $g$ as
measuring of how far
$T_{\lambda}$ is from being a symmetry of $S$.
Let us now consider
the variational problem associated to NLW on $\Omeg$ and
let $v$ be the vector field on $\RRn$ that generates the flow
$\Phi_{\lambda}$ through which $T_{\lambda}$ is defined.
Here the independent
variables are space-time variables $(x,\theta)\in\Omeg$ so that
$\n_{m}= \n_{x,\theta}\equiv(\n,{\cal D})$.
With $g$ as defined in (A.24), and noting that a
divergence
term vanishes when integrated over $\Omeg$
(viz. the term $\n_{m}\cdot {\cal S}v$), if $\gamma$
is a critical point of $S$, the
virial relation associated to the vector field $v$ (equation (A.13))
can be written as
\begin{equation}
S'[\gamma](v\cdot\n\gamma)\;=\;
\;\int_{\Omeg} g
\;\;=\;\;0.
\end{equation}
If in addition $\gamma$ solves the Euler-Lagrange equation, then
we can write $g$ as
\begin{eqnarray}
g &=&
({\cal S}_{u} - \n_{x,\theta}\cdot{\cal S}_{p})v\cdot\n\varphi
+ \n_{x,\theta}\cdot({\cal S}_{p}v\cdot\n\varphi -
{\cal S}v) \nonumber \\
& = &
\n_{x,\theta}\cdot({\cal S}_{p}v\cdot\n\varphi -
{\cal S}v) \nonumber \\
& = &
{\cal D}\cdot {\bf e} \;+\; \nabla\cdot
{\bf p}
\end{eqnarray}
where $ {\bf e} =
(\partial{\cal S}/\partial{\cal D}\gamma)v\cdot\n\gamma -
{\cal S}v_{b}$
and ${\bf p}= (\partial{\cal S}/\partial\n\gamma)v\cdot\n\gamma
- {\cal S}v_{a}$
(cf. (A.17)).
With the definitions
${\bf E}_{v}(\gamma) \equiv \int_{\RRn}{\bf e}$ and
$G\equiv\int_{\RR^{N}}g$, we see from (A.26) that
$G$ acts as source of ${\bf E}_{v}(\gamma)$;
\begin{equation}
{\cal D}\cdot{\bf E}_{v}(\gamma)\;=\;G.
\end{equation}
This corroborates the statement made above about $g$
measuring how far $T_{\lambda}$ is from being a symmetry group
of $S$:\ if $T_{\lambda}$ is a symmetry group of $S$ then
${\bf E}_{v}(\gamma)$ as defined through (A.26) is divergence free;
\ if
$T_{\lambda}$ is not a symmetry group of $S$ then
the divergence of ${\bf E}_{v}(\gamma)$ is determined by $g$.
\vspace{.2in}
For the Lagrangian associated to NLW on $\Omeg$ (equation (A.3)),
equation (A.24) is
\begin{equation}
g = tr\,dv\,\left(\frac{1}{2}\left(\Dw\gamma\right)^{2}
- F(\gamma)\right)
+ \n\gamma\cdot[dv - \frac{1}{2}tr\,dv\, {\bf 1}]\n\gamma
\end{equation}
which was derived in Theorem 2.7 (cf. (A.25)).
By Lemma A.2 this function vanishes if $T_{\lambda}$ is a
symmetry group of NLW. For example,
rotations and translations of $\RRn$ are symmetries of
NLW. In the former case $dv\in so(N)$ and in the latter $v=constant$.
In both cases it is apparent from (A.28) that $g = 0$.
\vspace{.2in}
As another application of the variational calculus, we
point out that the formula (A.28)
may be derived directly from (A.13) (cf. (A.25)) as follows.
First note that
$$
S[\gamma_{\lambda}]\;=\;\int_{\Omeg}\Big\{
-\frac{1}{2}(\Dw\gamma_{\lambda})^{2} + \frac{1}{2}|\n\gamma_{\lambda}
|^{2}
+ F(\gamma_{\lambda})\Big\}\,dxd\theta.
$$
By making the change of variables $y = \Phi_{\lambda}(x)$, this becomes
$$
S[\gamma_{\lambda}]\;=\;\int_{\Omeg}\Big\{
-\frac{1}{2}(\Dw\gamma)^{2}
+ \frac{1}{2}|\n\gamma_{\lambda}\circ\Phi_{-\lambda}|^{2}
+ F(\gamma)\Big\}
\,det\,(J_{\lambda}\circ\Phi_{-\lambda})^{-1}\,dyd\theta
$$
where $[J_{\lambda}(x)]_{i,j} = \partial\Phi^{i}_{\lambda}/
\partial x_{j}$ is the Jacobian associated to the transformation
$x\mapsto y$ and $(J_{\lambda}\circ\Phi_{-\lambda})^{-1}$ is the
Jacobian associated to the inverse mapping $y\mapsto x$.
Note that
$$
\frac{\partial}{\partial\lambda}\,det\,(J_{\lambda}\circ\Phi_{-
\lambda})^{-1}\Big|_{\lambda = 0}\;\;=\;\;tr\,dv.
$$
In addition,
$$
|\n\gamma_{\lambda}(x)|^{2}\;=\;\n\gamma_{\lambda}(x)\cdot
\n\gamma_{\lambda}(x)\;=\;A\n\gamma(y)\cdot\n\gamma(y),
\;\;\mbox{ where }\;[A]_{i,j} =\sum_{k=1}^{N}
\frac{\partial\Phi^{i}_{\lambda}}{\partial x_{k}}
\frac{\partial\Phi^{j}_{\lambda}}{\partial x_{k}},
$$
and
$$
\frac{\partial}{\partial\lambda}\Big[A\circ\Phi_{\lambda}\Big]
\Big|_{\lambda = 0}\;\;=\;\;dv + dv^{T},
$$
which together imply that
$$
\frac{\partial}{\partial\lambda}|\n\gamma_{\lambda}\circ\Phi_{-\lambda}
|^{2}\,\Big|_{\lambda = 0}\;\;=
\;\;2\,\n\gamma\cdot\,dv\,\n\gamma.
$$
Therefore,
\begin{eqnarray*}
\frac{d}{d\lambda}S[\gamma_{\lambda}]\Big|_{\lambda = 0}
&=& \int_{\Omeg}\frac{\partial}{\partial\lambda}
\Big\{
-\frac{1}{2}(\Dw\gamma)^{2} + \frac{1}{2}|\n\gamma_{\lambda}
\circ\Phi_{-\lambda}
|^{2} + F(\gamma)\Big\}
det\,(J_{\lambda}\circ\Phi_{-\lambda})^{-1}
\Big|_{\lambda = 0}\,dyd\theta \\
& = &
\int_{\Omeg} g
\end{eqnarray*}
with $g$ as in (A.28).
\newpage
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\noindent
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