Local Boundedness for Vector Valued Minimizers of Anisotropic Functionals
Francesco Leonetti, Elvira Mascolo · Zeitschrift für Analysis und ihre Anwendungen · 2012
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\limits_{j=1}^{n} g_j(x,|u_{x_j}|) and f(x,Du) = a(x, |(u_{x_1},\ldots ,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) .