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

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.