On symmetry of nonnegative solutions of elliptic equations

Poláčik, P.

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 1, pp. 1-19.

Nous considérons le problème de Dirichlet pour une classe dʼéquations elliptiques complètement non-linéaires sur un domaine borné Ω. Nous supposons que Ω est symétrique par rapport à un hyperplan H et convexe dans la direction perpendiculaire à H. Par un résultat bien connu de Gidas, Ni et Nirenberg ainsi que par ses généralisations, toutes les solutions positives sont symétriques par rapport à une réflextion de H et décroissent à partir de leur distance de lʼhyperplan dans la direction orthogonale à H. Pour les solutions non-négatives, ce résultat nʼest pas toujours vrai. Nous montrons que, néanmoins, le résultat sur la symétrie reste valable pour les solutions positives : toute solution non-négative u est symétrique par rapport à H. En outre, nous montrons que si $u\neg \equiv 0$, alors lʼensemble nodal de u divise le domaine Ω en un nombre fini de sous-domaines symétriques sous réflextion dans lesquels u possède la symétrie habituelle de Gidas–Ni–Nirenberg et des propriétés de monotonie. Nous montrons aussi plusieurs exemples de solutions positives avec un ensemble nodal intérieur non vide.

We consider the Dirichlet problem for a class of fully nonlinear elliptic equations on a bounded domain Ω. We assume that Ω is symmetric about a hyperplane H and convex in the direction perpendicular to H. By a well-known result of Gidas, Ni and Nirenberg and its generalizations, all positive solutions are reflectionally symmetric about H and decreasing away from the hyperplane in the direction orthogonal to H. For nonnegative solutions, this result is not always true. We show that, nonetheless, the symmetry part of the result remains valid for nonnegative solutions: any nonnegative solution u is symmetric about H. Moreover, we prove that if $u\neg \equiv 0$, then the nodal set of u divides the domain Ω into a finite number of reflectionally symmetric subdomains in which u has the usual Gidas–Ni–Nirenberg symmetry and monotonicity properties. We also show several examples of nonnegative solutions with a nonempty interior nodal set.

Classification : 35J60, 35B06, 35B05

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