On nodal domains in Euclidean balls
Résumé
A. Pleijel (1956) has proved that in the case of the Laplacian with Dirichlet condition, the equality in the Courant nodal theorem (Courant sharp situation) can only be true for a nite number of eigenvalues when the dimension is 2. Recently Polterovich extended the result to the Neumann problem in two dimensions in the case when the boundary is piecewise analytic. A question coming from the theory of spectral minimal partitions has motivated the analysis of the cases when one has equality in Courant's theorem. We identify the Courant sharp eigenvalues for the Dirichlet and the Neumann Laplacians in balls in R d , d 2. It is the rst result of this type holding in any dimension. The corresponding result for the Dirichlet Laplacian in the disc in R 2 was obtained by B. Heler, T. Homann-Ostenhof and S. Terracini. IF Introduction and main results e onsider the prolem of ounting nodl domins of eigenfuntions of the selfE djoint reliztion of the vplinD ¡ in the unit ll in R d F he nodl domins4 re the onneted omponents of the zeroset of the eigenfuntion in the llF e onsider the hirihlet prolem for d ! Q nd the xeumnn prolem for d ! P @the orresponding results for the hirihlet prolem for d a P ws given in UAF o e more preiseD denoting y n the nth eigenvlueD our gol is to disuss the property of gournt shrpness of these opertorsD tht is the existene of eigenvlE ues n for whih there exists n eigenfuntion with extly n nodl dominsF e rell tht gournt9s theorem sys tht the numer of nodl dominsD @©AD of n eigenfuntion © orresponding to n is ounded y nF woreoverD it hs een proven tht the numer of gournt shrp ses must e niteD see IU for the hirihlet se nd IW for the xeumnn se @in dimension P only nd for pieewise nlyti oundriesAF he two rst eigenvlues re lwys gournt shrpF e will prove the followingF Theorem 1.1. The only Courant sharp eigenvalues for the Neumann Laplacian for the disc are 1 , 2 and 4. Theorem 1.2. The only Courant sharp eigenvalues for the Dirichlet and Neumann Laplacians for the ball in R d , d ! Q, are 1 and 2. his nlysis is motivted y the prolem of spetrl miniml kEprtitionsD where one is interested in minimizing mx j 1 @D j A over the fmily h a @D 1 ; ¡ ¡ ¡ ; D k A of pirwise disjoint open sets in domin D where 1 @D j A denotes either the hirihlet ground stte energy @if we nlyze the hirihlet spetrl prtitions of n open set A or the hirihlet{xeumnn ground stte energy for the vplin in D j with xeumnn ondition on @D j @ nd hirihlet ondition on the remining prt of @D j F here re now mny results in the twoEdimensionl @PhA seF e refer to R 2010 Mathematics Subject Classication. 35B05; 35P20, 58J50.
Domaines
Mathématiques [math]
Origine : Fichiers produits par l'(les) auteur(s)
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