Regularity of a free boundary with application to the Pompeiu problem

Luis A. Caffarelli, Lavi Karp and Henrik Shahgholian

Review from Zentralblatt MATH:

The main part of the paper is devoted to the following free boundary value problem. Let $B$ be the unit ball in $\bbfR^N$ centered at the origin and look for a domain $\Omega\subset \bbfR^N$ such that $$\Delta u= \chi_\Omega\quad\text{in }B,\quad u=|\nabla u|= 0\quad\text{in }B\setminus \Omega,\quad 0\in

tial\Omega.$$ The aim of the paper is to find conditions ensuring the analyticity of $

tial\Omega$ near the origin. The desired result is obtained through three theorems (each one interesting in itself), that were previously known only in two dimensions.

Define the class $P_r(z,M)$ as the set of functions $u$ satisfying $$\Delta u= \chi_\Omega\quad\text{in }B_r(z),\quad u= |\nabla u|=0 \quad\text{in }B_r(z)\setminus \Omega,\quad \|u\|_{\infty|B_r(z)}\le M,\quad z\in

tial\Omega,$$ and the class $P_\infty(0,M)$ of the solutions in $\bbfR^N$ with growth condition $|u(x)|\le M(|x|^2+ 1)$. Define also (i) the minimal diameter MD$(D)$ of a bounded set $D$ as the infimum of distances between pairs of parallel planes bounding a strip containing $D$, (ii) the density function $$\delta_r(u)= {1\over r}\text{MD}\{(u= |\nabla u|= 0\cap B_r(0)\}.$$ The three basic theorems are the following:

Theorem 1. If $u\in P_1(z, M)$ there exists a constant $C$, depending only on the dimension $N$, such that $\sup_{B_{{1\over 2}}(z)}\|D_{ij}u\|\le CM$.

Theorem 2. Let $u\in P_\infty(0,M)$ and $\delta_r(u)> 0$ for some $r$. Then $u\ge 0$ in $\bbfR^N$ and $D_{ee} u\ge 0$ in $\Omega$ for any direction $e$ ($\Omega^c$ is convex). Moreover, if $\lim_{R\to\infty} \sup\delta_R(u)> 0$, then $u= {1\over 2}(\max(x_1, 0))^2$ in some coordinate system.

Theorem 3. If there exists a modulus of continuity $\sigma$ such that if $u\in P_1(D, M)$ and $\delta_{r_0}(u)> \sigma(r_0)$ for some $r_0< 1$, then $

tial\Omega$ is the graph of a $C^1$ function in $B(0, C_0, r^2_0)$, with $C_0= C_0(M, N)$.

As a Corollary to Theorem 3, results existing in the literature imply that the assumption of Theorem 3 guarantees the analyticity of $

tial\Omega$ near the origin.

Theorem 1 and 3 are extended to the non-general operator $\Sigma_{ij}D_i(a_{ij} D_ju)+ a(x)u$. In particular this implies that the same analyticity results holds when the Laplacian is replaced by the Helmholtz operator $\Delta u+u$ (the problem is then related to the so-called Pompeiu problem).

Reviewed by Antonio Fasano

Keywords: Laplace equation; Helmholtz equation; Pompeiu problem; analyticity of $\partial\Omega$ near the origin

Classification (MSC2000): 35R35 35J05 31B20

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