Schiffer's conjecture

General Description View the group

The Pompeiu problem states that if the integral of a nonzero continuous function on $\mathbb R^n$ vanishes on all congruent copies of a bounded Lipschitz domain $K$ then the domain is a ball. The problem can be reformulated in terms of PDEs: Suppose $v$ solves $$ \Delta v + \lambda v = \chi_{D}, \ \hbox{in }\mathbb R^n, \qquad v=0 \ \hbox{outside } D, $$ for some domain bounded domain $D$. Here $n\geq 2$. Is $D$ a ball?

The problem is connected to Schiffer's conjecture, and generally to free boundary problems.

See the video: https://www.youtube.com/watch?v=7nZgCBRZLds for some modified version of the conjecture and a finite element approach.

Problem's Description

Let $u$ solve $\Delta u+k^2u=0$ in $K$, with $u=0$, $|\nabla u|=1$, on $\partial K$, where the latter is a compact Lipschitz domain in $\mathbb R^3$ Then $K$ is a ball.

1. ArticleIs an originSur certains systèmes d'équations linéaires et sur une propriété intégrale des fonctions de plusieurs variables

   • D. Pompeiu

   Comptes Rendus de l'Académie des Sciences. Série I. Mathématique, 188, 1138-1139, 1929