Symmetry problem

Abstract

A novel approach to an old symmetry problem is developed. A new proof is given for the following symmetry problem, studied earlier: if Δ u = 1 \Delta u=1 in D ⊂ R 3 D\subset \mathbb {R}^3 , u = 0 u=0 on S S , the boundary of D D , and u N = c o n s t u_N=const on S S , then S S is a sphere. It is assumed that S S is a Lipschitz surface homeomorphic to a sphere. This result has been proved in different ways by various authors. Our proof is based on a simple new idea.