Symmetry results for solutions of equations involving zero-order operators

Disson dos Prazeres; Ying Wang

doi:10.1016/j.crma.2015.12.013

Partial differential equations

Symmetry results for solutions of equations involving zero-order operators

Presented by: the Editorial Board

Disson dos Prazeres ¹; Ying Wang 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, Volume 354 (2016) no. 3, pp. 277-281.

Abstract
Résumé

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.

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.

Received: 2015-10-04
Accepted: 2015-12-14
Published online: 2016-02-03

DOI: 10.1016/j.crma.2015.12.013

Author's affiliations:

Disson dos Prazeres ¹; Ying Wang 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

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     author = {Disson dos Prazeres and Ying Wang},
     title = {Symmetry results for solutions of equations involving zero-order operators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {277--281},
     publisher = {Elsevier},
     volume = {354},
     number = {3},
     year = {2016},
     doi = {10.1016/j.crma.2015.12.013},
     language = {en},
}

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Disson dos Prazeres; Ying Wang. Symmetry results for solutions of equations involving zero-order operators. Comptes Rendus. Mathématique, Volume 354 (2016) no. 3, pp. 277-281. doi : 10.1016/j.crma.2015.12.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.12.013/

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