Singular quasilinear elliptic equations and Hölder regularity

Giacomoni, Jacques; Schindler, Ian; Takáč, Peter

doi:10.1016/j.crma.2012.04.007

Partial Differential Equations

Singular quasilinear elliptic equations and Hölder regularity
[Régularité höldérienne pour des équations quasi-linéaires elliptiques singulières]

Présenté par : Haïm Brézis

Giacomoni, Jacques ¹ ; Schindler, Ian ² ; Takáč, Peter ³

¹ Laboratoire de mathématiques et de leurs applications - UMR CNRS 5142, bâtiment IPRA, université de Pau et des Pays de lʼAdour, avenue de lʼuniversité, BP 1155, 64013 Pau, France
² MIP-CEREMATH / bâtiment C, manufacture des tabacs, allée de Brienne 21, 31000 Toulouse, France
³ Institut für Mathematik, Universität Rostock, Ulmenstraße 69, Haus 3, 18055 Rostock, Germany

Comptes Rendus. Mathématique, Tome 350 (2012) no. 7-8, pp. 383-388.

Résumé
Abstract

Nous démontrons la régularité höldérienne (Théorème 2.1) des solutions faibles des équations quasi-linéaires elliptiques singulières de la forme suivante :

{\begin{matrix} - Δ_{p} u = \frac{K (x)}{u^{δ}} + g (x) & in Ω; \\ u |_{\partial Ω} = 0, u > 0 & in Ω, \end{matrix}

(P)

où Ω est un ouvert borné régulier,

1 < p < \infty

δ > 0

K \in L_{loc}^{\infty} (Ω)

satisfait

0 ⩽ K (x) ⩽ const \cdot dist {(x, \partial Ω)}^{- ω}

pour p.p.

x \in Ω

0 < ω < 1 + (1 - δ) (1 - \frac{1}{p})

0 ⩽ g \in L^{\infty} (Ω)

. Théorème 2.1 combiné avec le théorème du point fixe de Schauder permet de démontrer lʼexistence de solutions faibles de systèmes elliptiques quasi-linéaires singuliers de la forme (PS) (voir Introduction).

We prove the Hölder regularity (Theorem 2.1) for weak solutions to singular quasilinear elliptic equations whose prototype is

{\begin{matrix} - Δ_{p} u = \frac{K (x)}{u^{δ}} + g (x) & in Ω; \\ u |_{\partial Ω} = 0, u > 0 & in Ω, \end{matrix}

(P)

where Ω is an open bounded domain with smooth boundary,

1 < p < \infty

δ > 0

K \in L_{loc}^{\infty} (Ω)

satisfies

0 ⩽ K (x) ⩽ const \cdot dist {(x, \partial Ω)}^{- ω}

for a.e.

x \in Ω

0 < ω < 1 + (1 - δ) (1 - \frac{1}{p})

, and

0 ⩽ g \in L^{\infty} (Ω)

. Theorem 2.1 together with the Schauder fixed point theorem can be used to obtain the existence of weak solutions to the singular quasilinear elliptic system (PS) described in the Introduction.

Reçu le : 2012-02-06
Accepté le : 2012-04-05
Publié le : 2012-04-01

DOI : 10.1016/j.crma.2012.04.007

Affiliations des auteurs :

Giacomoni, Jacques ¹ ; Schindler, Ian ² ; Takáč, Peter ³

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     title = {Singular quasilinear elliptic equations and {H\"older} regularity},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {383--388},
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Giacomoni, Jacques; Schindler, Ian; Takáč, Peter. Singular quasilinear elliptic equations and Hölder regularity. Comptes Rendus. Mathématique, Tome 350 (2012) no. 7-8, pp. 383-388. doi : 10.1016/j.crma.2012.04.007. http://www.numdam.org/articles/10.1016/j.crma.2012.04.007/

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