Local Boundedness for Vector Valued Minimizers of Anisotropic Functionals

Abstract

For variational integrals $\mathcal{F}(u)={\int }_{\Omega }f(x,Du)dx$ defined on vector valued mappings $u:\Omega \subset {\mathbb{R}}^n\to {\mathbb{R}}^N$, we establish some structure conditions on $f$ that enable us to prove local boundedness for minimizers $u\in W^{1,1}(\Omega ;{\mathbb{R}}^N)$ of $\mathcal{F}$. These structure conditions are satisfied in three remarkable examples: $f(x,Du)=g(x,|Du|)$, $f(x,Du)=\sum _{j=1}^n{g}_j(x,|u_{x_j}|)$ and $f(x,Du)=a(x,|(u_{x_1},\dots ,u_{x_{n-1}})|)+b(x,|u_{x_n}|)$, for suitable convex functions $t\to g(x,t)$, $t\to g_j(x,t)$, $t\to a(x,t)$ and $t\to b(x,t)$.