Global regularity and stability of solutions to elliptic equations with nonstandard growth

Abstract

We study the regularity properties of solutions to elliptic equations similar to the p(·)-Laplacian. Our main results are a global reverse Hölder inequality, Hölder continuity up to the boundary and stability of solutions with respect to continuous perturbations in the variable growth exponent. We assume that the complement of the domain is uniformly fat in a capacitary sense. As technical tools, we derive a capacitary Sobolev–Poincaré inequality, and a version of Hardy's inequality.

Keywords:

• global higher integrability
• boundary regularity
• stability of solutions
• Sobolev–Poincaré inequality
• Hardy inequality

AMS Subject Classifications::

• 35J60
• 35B20
• 35J25
• 46E35

Acknowledgements

The first author wishes to thank the kind hospitality of J. Kinnunen and the Nonlinear PDE research group at the Institute of Mathematics (Helsinki University of Technology) for the nice and friendly atmosphere there. This project has been financially supported by the Academy of Finland and the Department of Mathematics of Trento. The third author is supported by the Norwegian Research Council project ‘Nonlinear Problems in Mathematical Analysis’.