JournalsaihpcVol. 33, No. 2pp. 451–476

Boundary regularity of minimizers of p(x)-energy functionals

  • Maria Alessandra Ragusa

    Dipartimento di Matematica e Informatica, Università di Catania, Viale Andrea Doria, 6-95125 Catania, Italy
  • Atsushi Tachikawa

    Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba, 278-8510, Japan
Boundary regularity of minimizers of p(x)-energy functionals cover
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Abstract

The paper is devoted to the study of the regularity on the boundary ∂Ω of a bounded open set ΩRm\Omega \subset \mathbb{R}^{m} for minimizers u for p(x)p(x)-energy functionals of the following type

E(u;Ω):=Ω(gαβ(x)Gij(u)Dαui(x)Dβuj(x))p(x)/2dx\mathscr{E}(u;\Omega ): = \int \limits_{\Omega }\left(g^{\alpha \beta }(x)G_{ij}(u)D_{\alpha }u^{i}(x)D_{\beta }u^{j}(x)\right)^{p(x)/ 2}dx

where (gαβ(x))(g^{\alpha \beta }(x)) and (Gij(u))(G_{ij}(u)) are symmetric positive definite matrices whose entries are continuous functions and p(x)2p(x) \geq 2 is a continuous function. The authors prove that such minimizers u have no singular points on the boundary.

Résumé

Dans cet article, les auteurs étudient la régularité sur la frontière ∂Ω d'un ouvert borné ΩRm\Omega \subset \mathbb{R}^{m} des minimiseurs u des fonctionnelles d'énergie p(x)p(x) du type suivant :

E(u;Ω):=Ω(gαβ(x)Gij(u)Dαui(x)Dβuj(x))p(x)/2dx,\mathscr{E}(u;\Omega ): = \int \limits_{\Omega }\left(g^{\alpha \beta }(x)G_{ij}(u)D_{\alpha }u^{i}(x)D_{\beta }u^{j}(x)\right)^{p(x)/ 2}dx,

(gαβ(x))(g^{\alpha \beta }(x)) et (Gij(u))(G_{ij}(u)) sont des matrices symétriques définies positives dont les éléments sont des fonctions continues et p(x)2p(x) \geq 2 est une fonction continue. Les auteurs prouvent que ces minimiseurs u n'ont pas de point singulier sur la frontière ∂Ω.

Cite this article

Maria Alessandra Ragusa, Atsushi Tachikawa, Boundary regularity of minimizers of p(x)-energy functionals. Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 2, pp. 451–476

DOI 10.1016/J.ANIHPC.2014.11.003