stability results for the N-dimensional Schiffer conjecture via a perturbation method

CANUTO BRUNO

CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS

SPRINGER

Lugar: Berlin; Año: 2014

0944-2669

In the present paper the author studies a overdeterminedeigenvalue problem, known in literature as Schiffer conjecture. Schifferconjecture can be formulated as follows: the only domain $Omega $ such thatthere exists a notrivial solution $arphi $ to the problem egin{equation}left{ egin{array}{rrrr}Delta arphi +mu arphi & = & 0 & mbox{in }Omega , \ partial _{{f n}}arphi & = & 0 & mbox{on }partial Omega ,end{array} ight.  label{s}end{equation}such that egin{equation}arphi =cmbox{ on }partial Omega ,  label{s´}end{equation}is the unit ball $B_1$ of ${Bbb {R}}^N$, $Ngeq 2$. Here $mu $ and $%arphi $ are respectively the eigenvalue and a corresponding eigenfunctionof $-Delta $ with Neumann boundary conditions. By assuming that $Omega $lies in a appropriate class of Lipschitz domains which are perturbations ofthe unit sphere $partial B_1$, we prove the following stability result ofthe Schiffer conjecture: egin{theorem}Given a $mu _{0m}$, for some $mgeq 1$, there exists a class ${cal D}$ of $%C^{2,alpha }$-domains, depending on $mu _{0m}$, such that if $arphi $ isa notrivial solution to ( ef{1}), ( ef{1´}), with $Omega in {cal D}$,and $mu =mu _{0m}^2+o(1)$, then $Omega =B_1$, $mu =mu _{0m}^2$, and $%arphi =I_0(mu _{0m}r)$ in $B_1$.end{theorem} oindent Here $I_0$ is the $N$-dimensional zero-order Bessel function offirst kind, and $mu _{0m}$ is the $m$-zero of the first order derivative of $I_0$, i.e. $I_0^{prime }(mu _{0m})=0$. oindent C. A. Berenstein [1] proves Schiffer conjecture by supposing thatthere exist infinitely many pairs $(mu _n,arphi _n)$ satisfying ( ef{s}%), ( ef{s´}) in ${Bbb R}^2$. This result has been extended for $Ngeq 3$by C. A. Berenstein and P. C. Yang [2].