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\begin{document}
{\centering
\bfseries
\Large
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
CRITICAL SETS OF SOLUTIONS TO ELLIPTIC EQUATIONS%
\footnote{Supported by Ministerium f\"ur Wissenschaft und Verkehr der
Republik \"Osterreich}%
\footnote{Work supported by the European Union TMR grant FMRX-CT 96-0001 }%
\footnote{Research partially supported by the NSF}\\[2\baselineskip]
\mdseries\scshape\small
R. Hardt$^1$\\
M. Hoffmann-Ostenhof$^2$ \\
T. Hoffmann-Ostenhof$^{3,4}$ \\
N. Nadirashvili$^5$\\[2\baselineskip]
\par
\upshape
Mathematics Department, Rice University, Houston$^1$\\
Institut f\"ur Mathematik, Universit\"at Wien$^2$\\
Institut f\"ur Theoretische Chemie, Universit\"at Wien$^3$\\
International Erwin Schr\"odinger Institute for Mathematical Physics$^4$\\
Department of Mathematics, MIT$^5$\\[2\baselineskip]
}
\begin{abstract}
Let $u\not\equiv\operatorname{const}$ satisfy an elliptic equation
$L_0u\equiv \sum a_{ij}D_{ij}u+\sum b_jD_ju=0$ with smooth coefficients
in a domain in $\mathbf R^n$. It is shown that the critical set
$|\nabla u|^{-1}\{0\}$ has locally finite $n-2$ dimensional Hausdorff
measure. This implies in particular that for a solution $u\not\equiv 0$
of $(L_0+c)u=0$, with $c\in C^\infty$, the critical zero set
$u^{-1}\{0\}\cap |\nabla u|^{-1}\{0\}$ has locally finite $n-2$ dimensional
Hausdorff measure.
\end{abstract}
\section*{1. Introduction and Main Results}
Let $\Omega$ be a domain in $\mathbf R^n$, $n\ge 3$ and let $u\not\equiv 0$
be a real-valued classical solution of the elliptic partial differential
equation
\begin{equation*}\tag{1.1}
{\cal L}u \equiv \sum_{i,j=1}^n a_{ij}D_{ij}u + \sum_{i=1}^n b_jD_ju+cu=0
\quad\text{in } \Omega
\end{equation*}
where the real valued coefficients $a_{ij}, b_j, c$ are $C^\infty$ functions
in $\Omega$.
We denote
\begin{equation*}
\Sigma(u) = |\nabla u|^{-1}\{0\}, \text{ and }\Sigma_0(u) =
\Sigma(u)\cap u^{-1}\{0\}.
\end{equation*}
In the following we shall show that locally the singular set $\Sigma_0$ of $u$
has finite $(n-2)$ - dimensional Hausdorff measure, i.e. $\mathcal H^{n-2}(\Sigma_0(u)\cap K)<\infty$ for all compact $K\subset\Omega$. The first author
and the remaining three authors independently wrote preprints proving
this result and the present paper is a combination of these two works.
For $n=2$, $\&(u)$ is well-known to consist of isolated points. For
$n\>3$ an elementary argument (see [HS], \sec 1.9), first given by
L. Caffarelli and A. Friedman [CF] for $\D u+f(x,u)=0$, shows that
$\&(u)$ is contained in a countable union of smooth $n-2$ dimensional
submanifolds. Q. Han [Hn] obtains similar structural results with much
weaker assumptions on the smoothness of the coefficients, In
particular, he proved that $\&(u)$ is essentially contained in a
countable union of $\c^{1,\a}$ graphs if the coefficients are
Lipschitz. But, even for smooth coefficients, the question remained
concerning the size of $\&(u)$. Last year it was shown in [HOHON] that for
$n=3$, $\Sigma_0(u)$ has locally finite 1-dimensional Hausdorff measure.
Here we generalize the result to $n\ge3$ dimensions. The result is obtained by
showing that the critical set $\Sigma$ of a solution of (1.1) with $c\equiv0$
has locally finite $n-2$ dimensional Hausdorff measure.
Recently there is a rather rich literature describing the `size' of the
zero set and in particular the singular set $\Sigma_0$ of solutions
to elliptic equations in terms of the appropriate Hausdorff measure,
respectively, Hausdorff dimension. See the list of references in the
introduction of [HOHON].
The size of the nodal set was considered in the conjecture of S.T. Yau
[Y] that $\H^{n-1}(u_\l^{-1}\{0\}) \sim c{\sqrt\l}$ for the
$\l$-eigenfunction $u_\l$ on a compact Riemannian manifold. This was
established for real analytic metrics by H. Donnelly and C. Fefferman
[DF1]. Note that, for real analytic coefficients, the local
finiteness, without estimates, of $\H^{n-1}(u^{-1}\{0\})$ (or of
$\H^{n-2}(\&(u))$) follows just from the real analyticity of $u$ [F],
3.4.8. For the nonanalytic case, R. Hardt and L. Simon [HS] proved
the local finiteness of $\H^{n-1}(u^{-1}\{0\})$ with the coefficients
being only Lipschitz smooth. However, for the Riemannian manifold
application, their upper estimate $C\l^{c\sqrt{\l}}$ is weaker than
Yau's conjecture. F.H. Lin and Q. Han [L], [HnL1], [HnL2] proved a
parabolic nodal set estimate (with time-independent coefficients),
simplified several arguments in [DF1] and [HS], and made estimates
involving the frequency (or order)
$N_R\==\[R\J_{\B_R}|\7u|^2\,dx\]/\[\J_{\6\B_R}u^2\,d\H^{n-1}]$. Lin
[L] also conjectured that
$$
\H^{n-1}(u^{-1}\{0\}\A\B_{R/2})\< CN_R\ \:and:\
\H^{n-2}(\&(u)\A\B_{R/2})\< CN_R^2\ .
$$
While more precise results are known in $2$ dimensions [AL, D, DF2, N],
the general Yau and Lin conjectures remain open. Two very recent preprints
give some nonexplicit bounds. [HHL1] treats coefficients with finite
differentiability, and [HHL2] treats higher order equations.
Basic for all these investigations is the asymptotic behaviour of $u(x)$ for
$x\rightarrow x_0$, where $u(x_0)=0$. Let ${\cal O}\in\Omega$, and let $u$ be a solution of (1.1), then it is well known (see e.g. [B]) that
\begin{equation*}\tag{1.2}
u(x)=p_M+O(|x|^{M+1})\text{ as }|x|\rightarrow0
\end{equation*}
where $p_M\not\equiv 0$ is a homogeneous polynomial of degree $M$ satisfying the
osculating equation
\begin{equation*}
\sum_{i,j}a_{ij}({\cal O})D_{ij}p_M=0.
\end{equation*}
Assume without loss of generality that $a_{ij}({\cal O})=\delta_{ij}$, then $p_M$ is
harmonic.
Therefore the investigations of the zero set respectively singular set of a
solution of (1.1) are motivated by the desire to understand to which extent
these sets can be described locally by the zero sets respectively critical
sets of
harmonic homogeneous polynomials. For a harmonic polynomial $P_M$ of degree $M$
in $n$ variables it is known (see e.g. [HS]) that for some $C(n)<\infty$
\begin{equation*}\tag{1.3}
\mathcal H^{n-2}(\Sigma(P_M)\cap B_1)\le C(n)M^2,
\end{equation*}
$B_1$ denoting a ball with radius $1$.
On the other hand there are examples showing that the singular
set of a solution of an elliptic equation can be rather wild. See [HOHON],
section 1. Conversations with L. Simon also led to the following simple
example: For any closed subset $K$ of $\mathbf R$, let $f$ be a nonnegative
smooth function vanishing exactly on $K$ with $|ff''|+|f'{}^2|<1/4$. Then
$u(x,y,z)=xy+f^2(z)$ satisfies the elliptic equation
$u_{xx}+u_{yy}+u_{zz}-(f^2)''(z)u_{xy}=0$, and has singular set
equaling $\{(0,0)\}\times K$.
To state now our main results we define the elliptic operator ${\cal L}_0$ by
\begin{equation*}
{\cal L}_0 = {\cal L} - c
\end{equation*}
with ${\cal L}$ and $c$ given according to (1.1).
{\bf Theorem 1.1.} {\it Let $u\not\equiv\operatorname{const}$ satisfy
\begin{equation*}\tag{1.1'}
{\cal L}_0u=0\quad\text{in }\Omega,\quad\Omega\subset\mathbf{R}^n.
\end{equation*}
Then for every compact subset $K$ of $\Omega$
\begin{equation*}
\mathcal H^{n-2}(\Sigma(u)\cap K) < \infty.
\end{equation*}}
{\bf Corollary 1.1.}{\it Let $u\not\equiv0$ satisfy equation (1.1). Then for every compact subset $K$ of $\Omega$
\begin{equation*}
\mathcal H^{n-2}(\Sigma_0(u)\cap K) < \infty.
\end{equation*}}
The Corollary is a rather immediate consequence of Theorem 1.1:
\textit{Proof of the Corollary.} Given $x_0\in\Omega$ there is a neighbourhood $U(x_0)$
and a $u_0\in C^{\infty}(U(x_0))$ with $u_0>0$ and ${\cal L}u_0 = 0$ in $U(x_0)$.
See e.g. [BJS], p.228. It is easily seen that $\mu\equiv uu_0^{-1}$ satisfies
in $U(x_0)$ an equation of type (1.1), so that by Theorem 1.1, $H^{n-2}(\Sigma(
\mu)\cap U') < \infty$ for every compact subset $U'$ of $U(x_0)$. Furthermore
the singular set of $u$ is a subset of the critical set of $\mu$.\qed
{\bf Remark:} That the assertion of Theorem 1.1 is false if ${\cal L}_0$ is
replaced by ${\cal L}$ can be seen from the following example: Let $v\in C^{
\infty}(B)$, $B\subset\mathbf{R}^n$, with $|v|<1$, then with $u=v^2+1$ and $c=(
\Delta v^2)(v^2+1)^{-1}$, $\Delta u+cu=0$ and $\Sigma(u)=v^{-1}\{0\}$. But
every closed subset of
$\mathbf R^n$ can be the zero set of a $C^\infty$-function (see e.g. [T])!
The structure of the proof of Theorem 1.1 is similar to the $3$-dimensional
case in [HOHON]. For this it was crucial to show (Theorem 3.1) that {\it in 3
dimensions, the complex dimension of the complex critical set of a
homogeneous real harmonic polynomial is at most one}. Here it is shown that
the complex critical set of a homogeneous harmonic polynomial $P$ with real
coefficients has at most complex dimension $n-2$ (Theorem 2.1). With this
result it can be proven that for suitable complex $2$-planes $\epsilon_{ij}$,
$1\le i1$ on $\R^n$ is a nonzero
function in the form
$$
u(x)\ =\ \sum_{\|\a\|=k}a_\a x^\a
$$
where $a_\a \in\R$ and $x^\a = x_1^{\a_1}\dots x_n^{\a_n}$ for
$x=(x_1,\dots,x_n)\in\R^n$, $\a=(\a_1,\dots,\a_n)\in\{0,1,\dots\}^n$, and
$\|\a\|\== \a_1+\dots +\a_n$.
The critical set $\&(v)$ of a polynomial $v(x)=\sum_{\|\a\|\\C$ is a complex {\it homogeneous} polynomial;
hence,
$$
\Sigma_{0\mathbf C}(p)=\Sigma_{\mathbf C}(p).
$$
For any complex 2-dimensional subspace $\epsilon\subset\C^n$, the restriction
$p|\epsilon$ is
essentially a complex homogeneous polynomial of two variables. Moreover,
{\it for $z\in\epsilon\sm\{0\}$,
$$
\7(p|_\epsilon)(z)=0
$$
if and only if either
$$
z\in\Sigma_{\mathbf C}(p)
$$
or}
$$
z\in p^{-1}\{0\}\sm\Sigma_\C(p)\ \ \:and:\ \ \epsilon\
\:is\ tangent\ to:\ p^{-1}\{0\}\ \:at:\ z\ . \;{3.1}
$$
For each pair $i,\, j$ of integers with $1\*\C^n$ also denote
the complex linear extension of $\g$. Thus each set
$(\p_{ij}\@\g)^{-1}\{y\}$, for $y\in\C^{n-2}$, is a complex affine $2$ plane
in $\C^n$.
\bb
\!{3.1 Lemma} {\it For any nonconstant homogeneous polynomial $p:\C^n\->\C$
having $\:dim:_\C\Sigma_\C(p)\ \<\ n-2$, there exists a rotation
$\g\in O(n)$ so that, for all integers $1\**0$ such that
\begin {equation*}\tag{4.3}
\operatorname{card}\Sigma(u)\cap (\pi_{ij}\circ\gamma)^{-1}\{y\}\cap B_R\le(M-1)^2
\end{equation*}
for all $y\in B_R^{(n-2)}$ and for all $i,j$ such that $1\le i0$, with
\begin {equation*}
\phi(y) = p(y) +o(|y|^k) \quad \text{for } |y|\rightarrow 0
\end{equation*}
and let $\phi _t(y)\in C^\infty(D({\cal O})\times I)$ for $t\in I$ where $I=
[-t_0,t_0]$, with $\phi_0 = \phi$. Then there exists $\tilde r$,
$0<\tilde r 0$
\begin{equation*}
\operatorname{card} (\Sigma (u|_{(\pi_{ij}\circ\gamma)^{-1}\{\sigma (t)\}})\cap D_{\tilde r}(\mathcal O))\le (M-1)^2
\end{equation*}
for $t$, $|t|\le t_0$, $t_0$ small enough. This implies further that for
some $\overline R >0$ and $\overline t >0$
\begin{equation*}\tag{4.4}
\operatorname{card}(\Sigma (u)\cap (\pi_{ij}\circ \gamma)^{-1}\{\sigma (t)\}\cap B_{\overline R})\le (M-1)^2\quad \forall t, |t|\le\overline t.
\end{equation*}
Suppose now for contradiction that Lemma 4.1 is false, then for some
$i,j$ there are sequences $\{R_k\}$ and $\{y^{(k)}\}$ with
$y^{(k)}\in\mathbf R^{n-2}$, $R_k\rightarrow 0$, $|y^{(k)}|\rightarrow 0$
for $k\rightarrow \infty$ such that
\begin{equation*}
\operatorname{card} (\Sigma (u)\cap(\pi_{ij}\circ\gamma)^{-1}\{y^{(k)}\}\cap B_{R_k})>(M-1)^2.
\end{equation*}
{\bf Proposition 4.3.} {\it
Let $\{y^{(k)}\}$ denote a sequence in $\mathbf R^n$ convergent to some
$\overline y$. Then there is a subsequence which is a subset of a
$C^\infty$-curve in $\mathbf R^n$.}
{\it Proof of Proposition 4.3:} We use a result of Kriegl [Kr] (see also Lemma
4.2.15 in [FK]):
Let $x_m\in\mathbf R^n$, $x_m\rightarrow\overline x$ for $m\rightarrow\infty$
and let $t_m\in\mathbf R$, $t_m\downarrow 0$ for $m\rightarrow\infty$. If
$\forall k$, $k\in\mathbf N$, $\{(x_m-x_{m+1})(t_m-t_{m+1})^{-k}\}$ is
bounded, then for some $C^\infty$-curve $\gamma$, $\gamma(t_m)=x_m$,
$\forall m$ and $\gamma^{(j)}(t_m)=0$, $\forall j\in \mathbf N$.
>From any convergent sequence $\{y^{(k)}\}$ it is easily seen that we can
pick a subsequence converging fast enough so that the assumptions above are
satisfied.\qed
Therefore we conclude: We can pick a subsequence of $\{y^{(k)}\}$
(again denoted by $\{y^{(k)}\}$) such that for some $\sigma\in C^\infty$,
$\sigma (t_k)=y^{(k)}$, $\forall k$ and
\begin{equation*}\tag{4.5}
\operatorname{card}(\Sigma (u)\cap (\pi_{ij}\circ\gamma)^{-1}\{\sigma (t_k)\}\cap B_{R_k})>(M-1)^2\quad \forall k.
\end{equation*}
On the other hand given $\sigma$, there are $\overline R,\overline t \>0$
such that (4.4) holds which is a contradiction to (4.5).
This verifies Lemma 4.1. \qed
\section*{5. Finiteness of the measure of the critical set.}
We first need
{\bf Lemma 5.1.}{\it Let $u\not\equiv\operatorname{const}$ satisfy (1.1') and $B$ be a ball with
$\overline B \subset \Omega$, then $\Sigma (u)\cap B$ decomposes into
to the countable union of subsets of a pairwise disjoint collection
of smooth $n-2$ dimensional submanifolds, i.e $\Sigma (u) \cap B$
is a countably $(n-2)$-rectifiable subset in the sense of Federer [F].}
{\it Proof of Lemma 5.1:} The proof is in principle the same as the
one of Lemma 1.9 in [HS]:
Thereby the argument is essentially that used by Cafarelli and Friedman
[CF]:
Let for $q=1,2,3,\dots $
\begin{equation*}
S_q=\{x|D^\alpha u(x)=0, \forall \alpha \text{ with } 0<|\alpha |\le q,
\quad D^{q+1}u(x)\ne 0\}.
\end{equation*}
Let $B_{2R}$ be a ball with $\overline B_{2R}\subset \Omega$, then
$\forall x_0\in \overline B_R$, $\mathcal L_0 (u-u(x_0))=0$.
Since the coefficients of the equation are smooth it follows via
unique continuation that $u-u(x_0)$ vanishes of finite order
$M(x_0)$ in $x_0$ and
\begin{equation*}
\sup_{x_0\in\overline B_{2R}}M(x_0)\equiv \overline M <\infty
\end{equation*}
Therefrom we obtain that $\forall a\in\Sigma(u)\cap B_R$
\begin{equation*}
B_R(a)\cap\{x|\nabla u(x)=0\} = B_R(a)\cap\bigcup _{q=1}^{\overline M}S_q.
\end{equation*}
The remaining part of the proof is the same as in e.g. [HS] or [CF].\qed
Due to Lemma 5.1 we have in particular
\begin{equation*}
\Sigma (u)\cap B_R =\cup_{m=1}^\infty E_m
\end{equation*}
where $E_1\subset E_2\subset\dots$ are Borel subsets of $\Sigma(u)$ of
finite $\mathcal H^{n-2}$-measure.
Without loss of generality we change coordinates to make $\gamma = \text{Id}$
in Lemma 4.1.
Then we use the integral geometric inequality 3.2.27 in [F] and Lemma 4.1 to
obtain the following estimate:
With $R>0$ given in Lemma 4.1
\begin{align*}
\mathcal H^{n-1}(\Sigma&(u)\cap \overline B_R)
=\lim_{m\rightarrow\infty}\mathcal H^{n-2}(E_m\cap\overline B_R)\\
&\le\limsup_{m\rightarrow\infty}\sum_{1\le i*