Partial Hölder continuity of minimizers of functionals satisfying a VMO condition
Christopher S. Goodrich · Advances in Calculus of Variations · 2015
Abstract For 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.