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

Giuffrè, Sofia ; Idone, Giovanna

Bollettino della Unione matematica italiana, Série 8, 8B (2005) no. 2, pp. 519-524

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 $1$, but $q$ close enough to 2, the solutions are global Hölder continuous for $n=2$.