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Article Dans Une Revue Proceedings of the American Mathematical Society Année : 2016

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 pro˜lem of ™ounting nod—l dom—ins of eigenfun™tions of the selfE —djoint re—liz—tion of the v—pl—™i—nD ¡ in the unit ˜—ll in R d F „he ’nod—l dom—ins4 —re the ™onne™ted ™omponents of the zeroset of the eigenfun™tion in the ˜—llF ‡e ™onsider the hiri™hlet pro˜lem for d ! Q —nd the xeum—nn pro˜lem for d ! P @the ™orresponding results for the hiri™hlet pro˜lem for d a P w—s given in ‘U“AF „o ˜e more pre™iseD denoting ˜y n the nth eigenv—lueD our go—l is to dis™uss the property of gour—nt sh—rpness of these oper—torsD th—t is the existen™e of eigenv—lE ues n for whi™h there exists —n eigenfun™tion with ex—™tly n nod—l dom—insF ‡e re™—ll th—t gour—nt9s theorem s—ys th—t the num˜er of nod—l dom—insD @©AD of —n eigenfun™tion © ™orresponding to n is ˜ounded ˜y nF woreoverD it h—s ˜een proven th—t the num˜er of gour—nt sh—rp ™—ses must ˜e niteD see ‘IU“ for the hiri™hlet ™—se —nd ‘IW“ for the xeum—nn ™—se @in dimension P only —nd for pie™ewise —n—lyti™ ˜ound—riesAF „he two rst eigenv—lues —re —lw—ys gour—nt sh—rpF ‡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 —n—lysis is motiv—ted ˜y the pro˜lem of spe™tr—l minim—l kEp—rtitionsD where one is interested in minimizing m—x j 1 @D j A over the f—mily h a @D 1 ; ¡ ¡ ¡ ; D k A of p—irwise disjoint open sets in — dom—in D where 1 @D j A denotes either the hiri™hlet ground st—te energy @if we —n—lyze the hiri™hlet spe™tr—l p—rtitions of —n open set A or the hiri™hlet{xeum—nn ground st—te energy for the v—pl—™i—n in D j with xeum—nn ™ondition on @D j ’ @ —nd hiri™hlet ™ondition on the rem—ining p—rt of @D j F „here —re now m—ny results in the twoEdimension—l @PhA ™—seF ‡e refer to ‘R“ 2010 Mathematics Subject Classication. 35B05; 35P20, 58J50.
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hal-01431631 , version 1 (11-01-2017)

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Bernard Helffer, Mikael Persson Sundqvist. On nodal domains in Euclidean balls. Proceedings of the American Mathematical Society, 2016, 144, pp.4777 - 4791. ⟨10.1090/proc/13098⟩. ⟨hal-01431631⟩
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