Global regularity for solutions to Dirichlet problem for discontinuous elliptic systems with nonlinearity $q>1$ and with natural growth

Giuffrè, Sofia; Idone, Giovanna

Global regularity for solutions to Dirichlet problem for discontinuous elliptic systems with nonlinearity

q > 1

and with natural growth

Giuffrè, Sofia ; Idone, Giovanna

Bollettino dell'Unione Matematica Italiana, Tome 8-A (2005), p. 519-524 / Harvested from Biblioteca Digitale Italiana di Matematica

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Abstract
Résumé

In this paper we deal with the Hölder regularity up to the boundary of the solutions to a nonhomogeneous Dirichlet problem for second order discontinuous elliptic systems with nonlinearity $q > 1$ and with natural growth. The aim of the paper is to clarify that the solutions of the above problem are always global Hölder continuous in the case of the dimension $n = q$ without any kind of regularity assumptions on the coefficients. As a consequence of this sharp result, the singular sets are always empty for $n = q$ .Moreover we show that also for $1 < q < 2$ , but $q$ close enough to 2, the solutions are global Hölder continuous for $n = 2$ .

In questo lavoro studiamo la regolarità fino alla frontiera di soluzioni di un problema di Dirichlet non omogeneo per sistemi ellittici discontinui del secondo ordine con non linearità $q > 1$ e con andamenti naturali. Scopo del lavoro è illustrare che le soluzioni del suddetto problema sono sempre globalmente hölderiane nel caso di dimensione $n = q$ senza alcun tipo di condizione di regolarità sui coefficienti. Come conseguenza di questo risultato, gli insiemi singolari $Ω_{0} \subset Ω$ , $Σ_{0} \subset \partial Ω$ sono sempre vuoti per $n = q$ . Inoltre dimostriamo che anche per $1 < q < 2$ , ma $q$ sufficientemente vicino a 2, le soluzioni sono globalmente hölderiane per $n = 2$ .

Publié le : 2005-06-01

MR 2149398 | Zbl 1150.35028

@article{BUMI_2005_8_8B_2_519_0,
     author = {Sofia Giuffr\`e and Giovanna Idone},
     title = {Global regularity for solutions to Dirichlet problem for discontinuous elliptic systems with nonlinearity $q>1$ and with natural growth},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {8-A},
     year = {2005},
     pages = {519-524},
     zbl = {1150.35028},
     mrnumber = {2149398},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2005_8_8B_2_519_0}
}

Giuffrè, Sofia; Idone, Giovanna. Global regularity for solutions to Dirichlet problem for discontinuous elliptic systems with nonlinearity $q>1$ and with natural growth. Bollettino dell'Unione Matematica Italiana, Tome 8-A (2005) pp. 519-524. http://gdmltest.u-ga.fr/item/BUMI_2005_8_8B_2_519_0/

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