Partial Hölder continuity of minimizers of functionals satisfying a VMO condition

Abstract

AbstractFor a bounded, open set${\Omega\hskip-0.569055pt\subseteq\hskip-0.569055pt\mathbb{R}^{n}}$we consider the partial regularity of vectorial minimizers${u\hskip-0.853583pt:\hskip-0.853583pt\Omega\hskip-0.853583pt\rightarrow\hskip-% 0.853583pt\mathbb{R}^{N}}$of the functional$u\mapsto\int_{\Omega}f(x,u,Du)\,dx,$where${f:\Omega\times\mathbb{R}^{N}\times\mathbb{R}^{N\times n}\rightarrow\mathbb{R}}$. The principal assumption we make is thatfis asymptotically related to a function of the form${(x,u,\xi)\mapsto a(x,u)F(\xi)}$, whereFpossessesp-Uhlenbeck structure and the partial maps${x\mapsto a(x,\cdot\,)}$and${u\mapsto a(\,\cdot\,,u)}$are, respectively, of class VMO and${\mathcal{C}^{0}}$. We demonstrate that any minimizer${u\in W^{1,p}(\Omega)}$of this functional is Hölder continuous on an open set${\Omega_{0}}$of full measure. Finally, we show by means of an example that our asymptotic relatedness condition is very general and permits a large class of functions.