H�lder continuity of local minimizers of vectorial integral functionals

Abstract

We study the regularity of vector-valued local minimizers in $ W^{1,p}, p > 1 $, of the integral functional $ u \mapsto \int_\Omega [(\mu^2 + |Du|^2)^{p/2} + f(x, u,|Du|)]dx, $ where Ω is an open set in $ \mathbb{R}^N $ and f is a continuous function, convex with respect to the last variable, such that $ 0 \leq f(x,u,t)\leq C(1+t^p) $. We prove that if f = f(x, t), or f = f(x, u, t) and $ p \leq N $, then local minimizers are locally Holder continuous for any exponent less than 1. If f = f(x, u, t) and p < N then local minimizers are Hoolder continuous for every exponent less than 1 in an open set $ \Omega_0 $ such that the Hausdorff dimension of $ \Omega \backslash \Omega_0 $ is less than N−p.