On the Local Everywhere Hölder Continuity of the Minima of a Class of Vectorial Integral Functionals of the Calculus of Variations

Abstract

In this paper we study the everywhere Hölder continuity of the minima of the following class of vectorial integral funcionals:

with some general conditions on the density \(G\).

We make the following assumptions about the function \(G\). Let \(\Omega\) be a bounded open subset of \(\mathbb{R}^{n}\), with \(n\geq 2\), and let \(G \colon \Omega \times\mathbb{R}^{m}\times\mathbb{R}_{0,+}^{m}\to \mathbb{R}\) be a Carathéodory function, where \(\mathbb{R}_{0,+}=[0,+\infty)\) and \(\mathbb{R} _{0,+}^{m}=\mathbb{R}_{0,+}\times \dots \times\mathbb{R}_{0,+}\) with \(m\geq 1\). We make the following growth conditions on \(G\): there exists a constant \(L>1\) such that

for \(\mathcal{L}^{n}\) a.e. \(x\in \Omega \), for every \(s^{\alpha}\in \mathbb{R}\) and every \(\xi^{\alpha}\in\mathbb{R}\) with \(\alpha=1,\dots,m\), \(m\geq 1\) and with \(a(x) \in L^{\sigma}(\Omega)\), \(a(x)\geq 0\) for \(\mathcal{L}^{n}\) a.e. \(x\in \Omega\), \(\sigma >{n}/{p}\), \(1\leq q<{p^{2}}/{n}\) and \(1<p<n\).

Assuming that the previous growth hypothesis holds, we prove the following regularity result. If \(u\,{\in}\, W^{1,p}(\Omega,\mathbb{R}^{m})\) is a local minimizer of the previous functional, then \(u^{\alpha}\in C_{\mathrm{loc}}^{o,\beta_{0}}(\Omega) \) for every \(\alpha=1,\dots,m\), with \(\beta_{0}\in (0,1) \). The regularity of minimizers is obtained by proving that each component stays in a suitable De Giorgi class and, from this, we conclude Hölder continuity.