Symmetry results for solutions of equations involving zero-order operators

dos Prazeres, Disson; Wang, Ying

doi:10.1016/j.crma.2015.12.013

Partial differential equations

Symmetry results for solutions of equations involving zero-order operators
[Symétries des solutions d'équations impliquant des opérateurs d'ordre zéro]

Présenté par : the Editorial Board

dos Prazeres, Disson ¹ ; Wang, Ying ²

¹ Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, UMR2071, CNRS-UChile, Universidad de Chile, Chile
² Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, PR China

Comptes Rendus. Mathématique, Tome 354 (2016) no. 3, pp. 277-281.

Résumé
Abstract

Soit $I_{ϵ} [u] (x) = \int_{R^{N}} \frac{u (y) - u (x)}{ϵ^{N + 2 σ} + | y - x |^{N + 2 σ}} d y$ , avec $ϵ > 0$ et $σ \in (0, 1)$ , un opérateur non local d'ordre zéro qui approche le laplacien fractionnaire lorsque ϵ tend vers 0. Nous étudions dans cette Note les symétries des solutions de l'équation (E) : $- I_{ϵ} [u] = f (u)$ dans la boule unité ouverte $B_{1}$ avec la condition $u = 0$ sur le complémentaire de la boule unité fermée. Nous observons que les propriétés de symétrie dépendent de la constante de Lipschitz de f. Lorsque cette constante de Lipschitz est majorée par $C_{N, σ} ϵ^{- 2 σ}$ , toute solution $u \in C ({\bar{B}}_{1})$ de (E) satisfaisant $u > c$ dans $B_{1}$ et $u = c$ sur le bord $\partial B_{1}$ est radialement symétrique.

In this note, we study symmetry results of solutions to equation (E) $- I_{ϵ} [u] = f (u)$ in $B_{1}$ with the condition $u = 0$ in ${\bar{B}}_{1}^{c}$ , where $I_{ϵ} [u] (x) = \int_{R^{N}} \frac{u (y) - u (x)}{ϵ^{N + 2 σ} + | y - x |^{N + 2 σ}} d y$ , with $ϵ > 0$ and $σ \in (0, 1)$ , is a zero-order nonlocal operator, which approaches the fractional Laplacian when $ϵ \to 0$ . The function f is locally Lipschitz continuous. We analyzed that the symmetry properties of solutions depend on the Lipschitz constant of f. When the Lipschitz constant is controlled by $C_{N, σ} ϵ^{- 2 σ}$ , any solution $u \in C ({\bar{B}}_{1})$ of (E) satisfying $u > c$ in $B_{1}$ and $u = c$ on $\partial B_{1}$ is radially symmetric.

Reçu le : 2015-10-04
Accepté le : 2015-12-14
Publié le : 2016-02-03

DOI : 10.1016/j.crma.2015.12.013

Affiliations des auteurs :

dos Prazeres, Disson ¹ ; Wang, Ying ²

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     title = {Symmetry results for solutions of equations involving zero-order operators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {277--281},
     publisher = {Elsevier},
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     year = {2016},
     doi = {10.1016/j.crma.2015.12.013},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2015.12.013/}
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dos Prazeres, Disson; Wang, Ying. Symmetry results for solutions of equations involving zero-order operators. Comptes Rendus. Mathématique, Tome 354 (2016) no. 3, pp. 277-281. doi : 10.1016/j.crma.2015.12.013. http://www.numdam.org/articles/10.1016/j.crma.2015.12.013/

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