Boundary regularity of minimizers of <i>p</i>(<i>x</i>)-energy functionals

Ragusa, Maria Alessandra; Tachikawa, Atsushi

doi:10.1016/j.anihpc.2014.11.003

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

Ragusa, Maria Alessandra ; Tachikawa, Atsushi

Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 2, pp. 451-476.

Résumé
Abstract

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

E (u; Ω) : = \int_{Ω} {(g^{α β} (x) G_{i j} (u) D_{α} u^{i} (x) D_{β} u^{j} (x))}^{p (x) / 2} d x,

où

(g^{α β} (x))

(G_{i j} (u))

sont des matrices symétriques définies positives dont les éléments sont des fonctions continues et

p (x) \geq 2

est une fonction continue. Les auteurs prouvent que ces minimiseurs u n'ont pas de point singulier sur la frontière ∂Ω.

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

E (u; Ω) : = \int_{Ω} {(g^{α β} (x) G_{i j} (u) D_{α} u^{i} (x) D_{β} u^{j} (x))}^{p (x) / 2} d x

where

(g^{α β} (x))

and

(G_{i j} (u))

are symmetric positive definite matrices whose entries are continuous functions and

p (x) \geq 2

is a continuous function. The authors prove that such minimizers u have no singular points on the boundary.

MR Zbl

DOI : 10.1016/j.anihpc.2014.11.003

Classification : 49N60, 35J50, 58E20
Mots clés : $ p(x)$-growth, Minimizer, Boundary regularity

@article{AIHPC_2016__33_2_451_0,
     author = {Ragusa, Maria Alessandra and Tachikawa, Atsushi},
     title = {Boundary regularity of minimizers of \protect\emph{p}(\protect\emph{x})-energy functionals},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {451--476},
     publisher = {Elsevier},
     volume = {33},
     number = {2},
     year = {2016},
     doi = {10.1016/j.anihpc.2014.11.003},
     zbl = {1333.49052},
     mrnumber = {3465382},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2014.11.003/}
}

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Ragusa, Maria Alessandra; Tachikawa, Atsushi. Boundary regularity of minimizers of p(x)-energy functionals. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 2, pp. 451-476. doi : 10.1016/j.anihpc.2014.11.003. http://www.numdam.org/articles/10.1016/j.anihpc.2014.11.003/

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