Polyconvex functionals and maximum principle

Author:

Carozza Menita1,Esposito Luca2,Giova Raffaella3,Leonetti Francesco4

Affiliation:

1. Department of Engineering, University of Sannio, Corso Garibaldi 107, 82100 Benevento, Italy

2. Department of Mathematics, University of Salerno, Via Ponte don Melillo Stecca 8, 84084 Fisciano (SA), Italy

3. Department of Economics and Law, University Parthenope of Napoli, Via Generale Parisi 13, 80132 Napoli, Italy

4. Department of Information Engineering, Computer Science and Mathematics, University of l'Aquila, Via Vetoio 67100 L'Aquila, Italy

Abstract

<abstract><p>Let us consider continuous minimizers $ u : \bar \Omega \subset \mathbb{R}^n \to \mathbb{R}^n $ of</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \mathcal{F}(v) = \int_{\Omega} [|Dv|^p \, + \, |{\rm det}\,Dv|^r] dx, $\end{document} </tex-math></disp-formula></p> <p>with $ p &gt; 1 $ and $ r &gt; 0 $; then it is known that every component $ u^\alpha $ of $ u = (u^1, ..., u^n) $ enjoys maximum principle: the set of interior points $ x $, for which the value $ u^\alpha(x) $ is greater than the supremum on the boundary, has null measure, that is, $ \mathcal{L}^n(\{ x \in \Omega: u^\alpha (x) &gt; \sup_{\partial \Omega} u^\alpha \}) = 0 $. If we change the structure of the functional, it might happen that the maximum principle fails, as in the case</p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \mathcal{F}(v) = \int_{\Omega}[\max\{(|Dv|^p - 1); 0 \} \, + \, |{\rm det}\,Dv|^r] dx, $\end{document} </tex-math></disp-formula></p> <p>with $ p &gt; 1 $ and $ r &gt; 0 $. Indeed, for a suitable boundary value, the set of the interior points $ x $, for which the value $ u^\alpha(x) $ is greater than the supremum on the boundary, has a positive measure, that is $ \mathcal{L}^n(\{ x \in \Omega: u^\alpha (x) &gt; \sup_{\partial \Omega} u^\alpha \}) &gt; 0 $. In this paper we show that the measure of the image of these bad points is zero, that is $ \mathcal{L}^n(u(\{ x \in \Omega: u^\alpha (x) &gt; \sup_{\partial \Omega} u^\alpha \})) = 0 $, provided $ p &gt; n $. This is a particular case of a more general theorem.</p></abstract>

Publisher

American Institute of Mathematical Sciences (AIMS)

Subject

Applied Mathematics,Mathematical Physics,Analysis

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