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Zeitschrift für Analysis und ihre Anwendungen


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Volume 31, Issue 3, 2012, pp. 357–378
DOI: 10.4171/ZAA/1464

Published online: 2012-07-11

Local Boundedness for Vector Valued Minimizers of Anisotropic Functionals

Francesco Leonetti[1] and Elvira Mascolo[2]

(1) Università degli Studi dell'Aquila, Italy
(2) Università di Firenze, Italy

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 $t \to b(x,t)$.

Keywords: Regularity, boundedness, minimizer, variational integral, elliptic system

Leonetti Francesco, Mascolo Elvira: Local Boundedness for Vector Valued Minimizers of Anisotropic Functionals. Z. Anal. Anwend. 31 (2012), 357-378. doi: 10.4171/ZAA/1464