Calculus of Variations and Geometric Measure Theory

G. Cupini - P. Marcellini - E. Mascolo

Local boundedness of solutions to quasilinear elliptic systems

created by cupini on 03 Sep 2012


Published Paper

Inserted: 3 sep 2012
Last Updated: 3 sep 2012

Journal: Manuscripta Mathematica
Volume: 137
Pages: 287-315
Year: 2012


The mathematical analysis to achieve everywhere regularity in the interior of weak solutions to nonlinear elliptic systems usually starts from their local boundedness. Having in mind De Giorgi's counterexamples, some structure conditions must be imposed to treat systems of partial differential equations. On the contrary, in the scalar case of a general elliptic single equation a well established theory of regularity exists. In this paper we propose a unified approach to local boundedness of weak solutions to a class of quasilinear elliptic systems, with a structure condition inspired by Ladyzhenskaya-Ural'tseva's work for linear systems, as well as valid for the general scalar case. Our growth assumptions on the nonlinear quantities involved are new and general enough to include anisotropic systems with sharp exponents and the $p,q-$growth case.

Keywords: systems of partial differential equations, local boundedness, $L^{\infty}$ regularity