JournalszaaVol. 31, No. 3pp. 357–378

Local Boundedness for Vector Valued Minimizers of Anisotropic Functionals

  • Francesco Leonetti

    Università degli Studi dell'Aquila, Italy
  • Elvira Mascolo

    Università di Firenze, Italy
Local Boundedness for Vector Valued Minimizers of Anisotropic Functionals cover
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Abstract

For variational integrals F(u)=Ωf(x,Du)dx\mathcal{F}(u)= \int_{\Omega} f(x,Du) \,dx defined on vector valued mappings u:ΩRnRNu:\Omega \subset \mathbb{R}^n \to \mathbb{R}^N, we establish some structure conditions on ff that enable us to prove local boundedness for minimizers uW1,1(Ω;RN)u \in W^{1,1}(\Omega;\mathbb{R}^N) of F\mathcal{F}. These structure conditions are satisfied in three remarkable examples: f(x,Du)=g(x,Du)f(x,Du)=g(x,|Du|), f(x,Du)=j=1ngj(x,uxj)f(x,Du) = \sum\limits_{j=1}^{n} g_j(x,|u_{x_j}|) and f(x,Du)=a(x,(ux1,,uxn1))+b(x,uxn)f(x,Du) = a(x, |(u_{x_1},\ldots ,u_{x_{n-1}})|) + b(x,|u_{x_n}|), for suitable convex functions tg(x,t)t \to g(x,t), tgj(x,t)t \to g_j(x,t), ta(x,t)t \to a(x,t) and tb(x,t)t \to b(x,t).

Cite this article

Francesco Leonetti, Elvira Mascolo, Local Boundedness for Vector Valued Minimizers of Anisotropic Functionals. Z. Anal. Anwend. 31 (2012), no. 3, pp. 357–378

DOI 10.4171/ZAA/1464