Symmetry problems for the Helmholtz equation

Abstract

Let D be a bounded domain in R2 with a smooth boundary S, N be the unit normal to S pointing out of D, k>0 is a constant.The class of over-determined problems of the type: ∇2u+k2u=c0inD,u|S=c1,uN|S=c2, is studied. Here uN|S is the normal derivative of u on S and cj, j=0,1,2, are constants.This problem was not studied as far as the author knows. Our result is the following theorem.Theorem.If|c1−c0k2|+|c2|>0and the above problem has a solution thenDis a ball.This theorem contains several earlier results, proved by the author, including the refined Schiffer’s conjecture and the refined Pompeiu problem.This paper continues the author’s investigations of symmetry problems for partial differential equations.