Abstract
In this paper we study the everywhere Hölder continuity of the minima of the following class of vectorial integral funcionals: \(\int_{\Omega}\sum_{\alpha=1}^{m}|\nabla u^{\alpha}|^{p} +G\bigl(x,u,|\nabla u^{1}|,\dots,|\nabla u^{m}|\bigr) \,dx,\) 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 \(\begin{aligned} \, \sum_{\alpha=1}^{m}|\xi^{\alpha}|^{q} -\sum_{\alpha=1}^{m}|s^{\alpha}|^{q}-a(x) &\leq G\bigl(x,s^{1},\dots,s^{m},|\xi^{1}|,\dots,|\xi^{m}|\bigr) \\ &\leq L\biggl[\sum_{\alpha=1}^{m}|\xi^{\alpha}|^{q}+\sum_{\alpha=1}^{m}|s^{\alpha }|^{q}+a(x) \biggr] \end{aligned}\) 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.
