Partial regularity of minimizers of asymptotically convex functionals with ${p(x)}$-growth
Volume 264 / 2022
Studia Mathematica 264 (2022), 71-102
MSC: Primary 46E35, 49N60; Secondary 35B65.
DOI: 10.4064/sm210104-20-9
Published online: 17 January 2022
Abstract
We consider vectorial minimizers of the integral functional \[ \int _{\Omega }f(x,u,Du)\, dx, \] where the function $(x,u,\xi )\mapsto f(x,u,\xi )$ is asymptotically related to a simpler function $(x,u,\xi )\mapsto a(x,u)|\xi |^{p(x)}$. Thus, we consider asymptotically convex integral functionals in the $p(x)$-growth setting. We demonstrate that minimizers are almost everywhere Hölder continuous, in a manner that mimics that simpler $p$-growth setting.
