\input amstex
\documentstyle{amsppt}
\def\Q{\nabla}
\def\k{\kappa}
\def\la{\lambda}
\topmatter
\title ON THE SECOND EIGENVALUE OF THE LAPLACE OPERATOR PENALIZED BY
CURVATURE\endtitle \rightheadtext{ON THE SECOND EIGENVALUE OF THE LAPLACE
OPERATOR}
\author EVANS M. HARRELL II\endauthor
\address GEORGIA INSTITUTE OF TECHNOLOGY\endaddress \email
harrell\@math.gatech.edu\endemail
\abstract
Consider the operator $-\Q^2 - q(\k)$, where $-\Q^2$ is the (positive)
Laplace-Beltrami operator on a closed manifold of the topological type of
the
two-sphere $S^2$ and $q$ is a symmetric non-negative quadratic form in the
principal
curvatures. Generalizing a well-known theorem of J.\ Hersch for the
Laplace-Beltrami operator alone, it is shown in this note that the second
eigenvalue $\la_1$ is uniquely maximized, among manifolds of fixed area, by
the
true sphere.
\endabstract
\endtopmatter
\document
Dimensionally, the Laplace operator $-\Q^2$ is comparable to the square of
curvature, both
having dimensions (length)$^{-2}$. Thus one might expect to encounter
partial differential
operators of the form $-\Q^2 - q$ in applications, where $q$ is a quadratic
expression in the
principal curvatures.
This was recently the case when N.\ Alikakos and G.\ Fusco performed a
stability analysis
of the interfacial surface separating two phases in one of the simpler
reaction-diffusion
models, the Allen-Cahn system. It was already realized in the first article
about this model
[5] that it exhibits interfaces moving according mean-curvature, as a
consequence of which
the model has attractive geometric features; for current state of
mathematical knowledge of
this see [9]. While simplified in comparison to most realistic
reaction-diffusion systems,
Allen-Cahn is a reasonable model at least for bistable alloys of iron and
aluminum. Picture
a bubble of material of phase I in a background of phase II. It undergoes
slow motions and
deformations, and if it is not at an external boundary, the surface
$\Omega$ smooths out and
eventually becomes round. According to Alikakos and Fusco, instabilities of
the surface
are associated with negative eigenvalues of an operator emerging from
linearization at $\Omega$ of
the form
$$-\Q^2-\sum^2_{j=1} \k^2_j,\tag1$$
where $\k_j$ are the principal curvatures at any given point of $\Omega$,
and $\Q^2$
is the Laplace-Beltrami operator on $\Omega$.
This can be thought of as a geometric Schr\"odinger operator with a
negative potential determined by curvature. It is evident that (1) is a
highly symmetric,
reasonable object, and that the second eigenvalue is special because it
equals 0 when $\Omega$
attains its target shape of a sphere. (While the analysis by Alikakos and
Fusco is not
accessible in print at the time of this writing, the lower dimensional
analogy is worked out
in the recent thesis of V.\ Stephanopoulos [12, see Proposition 5.4 and
Theorem 7.1; see
also 2]. Related work and an entry point for the literature on Allen-Cahn
and similar
reaction-diffusion models can be found in [1-3, 9].)
The conjecture was that for any other shape of $\Omega$ the second
eigenvalue of (1) is strictly
negative, and this is a special case of the theorem proved below. The
conjecture calls to
mind the theorem of J.\ Hersch [8] for the Laplace operator without the
curvature penalty,
whereby the unique such $\Omega$ maximizing the second eigenvalue is the
sphere. For the
Laplace operator plain and simple, the lowest eigenvalue is trivial, so the
second eigenvalue
is often referred to as the ``first." Unambiguously, the eigenvalues will
be written here $\la_0
< \la_1\le\dots$, and the one at issue is $\la_1$. Actually, Hersch's
problem was a bit more general, since he was concerned with
$$\frac{1}{\la_1} + \frac{1}{\la_2} + \frac{1}{\la_3},\tag2$$ and he also
allowed arbitrary weights on $\Omega$. The variational principle he used as
a lower
bound for (2) is not available for (1), because the latter is not positive
or even bounded
from below {\it a priori}. Hersch's technique is particular to two
dimensions, since it relies on
the ability to map a generic $\Omega$ conformally to $S^2$, among other
special facts, and in more
than two dimensions the conformal equivalence class is not large enough to
do this.
Moreover, the natural extension of Hersch's theorem to higher dimensions
has been shown
to be false by H.\ Urakawa [14]. As for Hersch, the operative definition of
the topological
type of the sphere used here will be conformal equivalence, so the theorem
of this article is
likewise restricted to two dimensions. More specifically, the following
lemma from [8]
will be needed:
\proclaim{Lemma}
(J. Hersch). Let $\Omega$ be a two-dimensional, closed, smooth Riemannian
manifold
of the topological type of the sphere, and specify a bounded, positive,
measurable function
$\rho$ on $\Omega$.
Then there exists a conformal transformation $\Phi: \Omega\to S^2\subset R^3$,
embedded in the
standard way as the unit sphere, such that $$\int_{S^2} {\bold x}\rho
(\Phi^{-1}(\bold x))Jd\hat S={\bold 0}.\tag 3$$
\endproclaim
Here ${\bold x} = (x,y,z)$ is the position vector in $R^3$,
$J$ is the Jacobian factor of the transformation, and $d\hat S$ is the
standard area element on $S^2$. Thinking of $\rho$
as a mass distribution, the
statement means that the center of mass is mapped to the origin in $R^3$.
Recall for later purposes that the restrictions of the Cartesian coordinate
functions $(x,y,z)$ to
the unit sphere are the spherical harmonics, which are the eigenfunctions
of $-\Q^2$ associated
with its (multiple) second eigenvalue [10]. Following Hersch's notation,
let $X,Y,Z$ denote
the functions on $\Omega$ obtained by mapping $(x,y,z)$ back to $\Omega$
with $\Phi^{-1}$.
Thus
$$X^2 + Y^2 + Z^2 = 1,\tag4$$
and (3) implies that $X$, $Y$, and $Z$ are orthogonal to $\rho$ on the
Hilbert space $L^2(\Omega)$ (endowed with the measure $dS$ arising from the
metric on $\Omega$).
\proclaim{Theorem}
Let $\k_{1,2}(p)$ denote the principal curvatures at the point
$p\in\Omega$, a closed manifold
of the topological type of $S^2$. Let $q(\xi,\eta)$ be a nonnegative
quadratic polynomial in
$\xi,\eta$,
such that $q(\eta,\xi) = q(\xi,\eta)$, and let $\la_0 < \la_1 \le\dots$
denote the eigenvalues of
$$ -\Q^2 - q(\k_1,\k_2)$$
as an operator on $L^2(\Omega)$. Then
$$\la_1\le\frac{4\pi (2-q(1,1)+q(0,0))}{|\Omega|}-q(0,0).\tag5$$ Equality
is attained if and only if $\Omega = r S^2$ (sphere of radius $r$).
\endproclaim
A plausibility argument for the theorem is to recall Hersch's theorem and
observe that the
quadratic form $q$ is minimized on average by the sphere. The obstacle to a
rigorous
argument along these lines is that since $\la_1$ is not the lowest
eigenvalue, it is characterized
by a min-max principle, and one must cope with the orthogonalization to the
ground-state
eigenfunction $u_0$ for $\la_0$. For a general $\Omega$ this $u_0$ is not
accessible.
\demo{Proof} Without loss of generality, subtract a constant from $q$ so
that $q(0,0) = 0$.
Inequality (5) then becomes
$$\la_1\le\frac{4\pi (2-q(1,1))}{|\Omega|}\tag6$$ Note that equality is
attained if $\Omega = r S^2$. Inequality (6) will follow if a function
$\zeta$ can be
exhibited, which is orthogonal to $u_0$, and for which the Rayleigh
quotient for this operator,
$$R(\zeta):=\frac{\int_\Omega|\Q\zeta|^2dS-\int_\Omega
q(\k_1,\k_2)|\zeta|^2dS} {\int_\Omega|\zeta|^2dS}\tag7$$
is less than the stated bound. As constructed, the functions $X$, $Y$, and
$Z$ are suitable
candidates for the Rayleigh quotient for $\la_1$ if $\rho$ is identified
with $u_0$, which is always
positive [7, Theorem 4.2.1; 11].
Next, recall that the Dirichlet integral in the numerator of (7) is
conformally invariant, i.e.,
$$
\int_\Omega|\Q X|^2dS=\int_{S^2}|\Q x|^2d\hat S=\frac{8\pi}{3}\tag8$$
(because $x$ is an eigenfunction for $-\Q^2$ on $S^2$ with eigenvalue 2).
With (8) and setting
$\zeta =X$ in (7), for example:
$$
R(X)=\frac{\frac{8\pi}{3}-\int_\Omega q(\k_1,\k_2)X^2dS}{\int_\Omega
X^2dS}$$ and similarly for $R(y)$ and $R(z)$.
Now, it is an elementary fact that if $c_j$ are positive numbers and for
each $j$,
$$a\le\frac{b_j}{c_j}$$
then
$$a\le\frac{\sum_j b_j}{\sum_jc_j}$$
(multiply by $c_j$ and sum). Thus, with (4),
$$\min(R_X,R_Y,R_Z)\le\frac{8\pi-\int_\Omega qdS}{\int_\Omega 1dS}.$$ A
simple calculation (writing $q(\xi,\eta) = a\xi\eta + b(\xi^2 + \eta^2)$
and completing the square) shows
that
$$q(\xi,\eta)\ge q(1,1) \xi\eta,\tag9$$
with equality only for $\xi=\eta$. Hence:
$$\min(R_X,R_Y,R_Z)\le\frac{8\pi-\int_\Omega q(1,1)\k_1,\k_2dS}{|\Omega|}
=\frac{4\pi (2-q(1,1))}{|\Omega|}$$
by the Gau\ss-Bonnet theorem, establishing (6). Because equality in (9)
requires $\xi=\eta$,
equality in (6) requires $\k_1=\k_2$ a.e., and by a theorem of Liebmann
[13, p.\ 122] this
characterizes $\Omega$ as the sphere $r S^2$. \hfill QED
\enddemo
As remarked above, a simple extension to more than two dimensions is
unlikely, but
related theorems probably apply to a) surfaces of higher genus as in [15],
b) surfaces with
boundaries which are predetermined, and c) one dimension. These cases all occur
physically and are under investigation [4]. An interface may meet the
predetermined edge
of a piece of alloy, producing case b) for example; while the
one-dimensional situation of a
curve describes an interface in a thin sheet of metal. Curiously, even the
one-dimensional
analogue is more complicated than this two-dimensional theorem, due to the
lack of
conformal invariance of the Dirichlet integral.
{\bf Acknowledgments.} I am grateful to N.\ Alikakos, I.\ Chavel, and J.\
Fleckinger for
helpful conversations and references, and to the Erwin Schr\"odinger
Institut in Vienna,
where part of this work was done. This research was supported by NSF grant
DMS-9211624.
\Refs
\ref\no1 \by Nicholas D. Alikakos, Peter W. Bates, and Xinfu Chen \paper
Convergence of the Cahn-
Hilliard equation to the Hele-Shaw model \paperinfo preprint\endref
\ref\no2 \by Nicholas D. Alikakos and Giorgio Fusco \paper The spectrum of
the Cahn-Hilliard operator for generic interface in higher space dimensions
\jour Indiana U. Math. J. \vol 4\yr 1993\pages 637-674\endref
\ref\no3 \by Nicholas D. Alikakos and Giorgio Fusco \paper Slow dynamics
for the Cahn-Hilliard equation in higher space dimensions. Part I: Spectral
estimates\jour Commun. in PDE
\toappear\endref
\ref\no4 \by Nicholas D. Alikakos, Giorgio Fusco, and Evans M. Harrell II
\paperinfo work in progress\endref
\ref\no5\by Samuel M. Allen and John W. Cahn\paper A microscopic theory for
antiphase boundary
motion and its application to antiphase domain coarsening\jour Acta Metall.
\vol 27
\yr 1979\pages1084-1095\endref
\ref\no6\by Shiu-Yuen Cheng
\paper A characterization of the 2-sphere by eigenfunctions\jour Proc. Amer.
Math. Soc. \vol 55\yr 1976\pages 379-381\endref
\ref\no7 \by Davies, E.B.
\paper Heat Kernels and Spectral Theory \jour Cambridge Tracts in
Mathematics \vol 92\publ Cambridge University Press\publaddr Cambridge\yr
1989\endref \ref\no8\by Joseph Hersch,
\paper Quatre propri\'et\'es isop\'erimetriques de membranes sph\'eriques
homog\`enes\jour C.R. Acad. Sci. Paris, s\'er A-B \vol 270\yr 1970\pages
A1645-1648\endref
\ref\no9\by Tom Ilmanen
\paper Convergence of the Allen-Cahn equation to Brakke's motion by mean
curvature\jour J. Diff. Geom. \vol 38\yr 1993\pages 417-461\endref
\ref\no10\by Klaus M\"uller\paper Spherical Harmonics \book Springer
Lecture Notes In Mathematics 17 \publ Springer-Verlag\publaddr Berlin\yr
1966\endref
\ref\no11 \by Michael Reed and Barry Simon\book Analysis of Operators,
Methods of Modern
Mathematical Physics IV\publ Academic\publaddr New York\yr 1978\endref
\ref\no12\by Vagelis Stephanopoulos\paper The role of critical eigenvalues
in a class of singularly
perturbed problems\paperinfo University of Tennessee dissertation,
December\yr 1993\endref
\ref\no13\by Dirk J. Struik
\book Lectures on Classical Differential Geometry\publ Dover\publaddr New
York\yr 1961\endref
\ref\no14\by Hajime Urakawa\paper
On the least positive eigenvalue of the Laplacian for compact group
manifolds\jour J. Math. Soc. Japan \vol 31\yr 1979\pages 209-226\endref
\ref\no15 \by Paul Yang and Shing-Tung Yau\paper
Eigenvalues of the Laplacian of compact Riemann surfaces and minimal
submanifolds\jour Ann Scuola Norm. Sup. Pisa, cl. sci (4)
\vol 17\yr 1980\pages 55-63\endref
\endRefs
Mathematics subject classification: 58G25.
\enddocument