### Mathematics in Engineering

2023, Issue 4: 1-10. doi: 10.3934/mine.2023077
Research article Special Issues

# Polyconvex functionals and maximum principle

• Received: 01 August 2022 Revised: 27 January 2023 Accepted: 14 February 2023 Published: 09 March 2023
• Let us consider continuous minimizers $u : \bar \Omega \subset \mathbb{R}^n \to \mathbb{R}^n$ of

$\mathcal{F}(v) = \int_{\Omega} [|Dv|^p \, + \, |{\rm det}\,Dv|^r] dx,$

with $p > 1$ and $r > 0$; 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, $\mathcal{L}^n(\{ x \in \Omega: u^\alpha (x) > \sup_{\partial \Omega} u^\alpha \}) = 0$. If we change the structure of the functional, it might happen that the maximum principle fails, as in the case

$\mathcal{F}(v) = \int_{\Omega}[\max\{(|Dv|^p - 1); 0 \} \, + \, |{\rm det}\,Dv|^r] dx,$

with $p > 1$ and $r > 0$. 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 $\mathcal{L}^n(\{ x \in \Omega: u^\alpha (x) > \sup_{\partial \Omega} u^\alpha \}) > 0$. In this paper we show that the measure of the image of these bad points is zero, that is $\mathcal{L}^n(u(\{ x \in \Omega: u^\alpha (x) > \sup_{\partial \Omega} u^\alpha \})) = 0$, provided $p > n$. This is a particular case of a more general theorem.

Citation: Menita Carozza, Luca Esposito, Raffaella Giova, Francesco Leonetti. Polyconvex functionals and maximum principle[J]. Mathematics in Engineering, 2023, 5(4): 1-10. doi: 10.3934/mine.2023077

### Related Papers:

• Let us consider continuous minimizers $u : \bar \Omega \subset \mathbb{R}^n \to \mathbb{R}^n$ of

$\mathcal{F}(v) = \int_{\Omega} [|Dv|^p \, + \, |{\rm det}\,Dv|^r] dx,$

with $p > 1$ and $r > 0$; 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, $\mathcal{L}^n(\{ x \in \Omega: u^\alpha (x) > \sup_{\partial \Omega} u^\alpha \}) = 0$. If we change the structure of the functional, it might happen that the maximum principle fails, as in the case

$\mathcal{F}(v) = \int_{\Omega}[\max\{(|Dv|^p - 1); 0 \} \, + \, |{\rm det}\,Dv|^r] dx,$

with $p > 1$ and $r > 0$. 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 $\mathcal{L}^n(\{ x \in \Omega: u^\alpha (x) > \sup_{\partial \Omega} u^\alpha \}) > 0$. In this paper we show that the measure of the image of these bad points is zero, that is $\mathcal{L}^n(u(\{ x \in \Omega: u^\alpha (x) > \sup_{\partial \Omega} u^\alpha \})) = 0$, provided $p > n$. This is a particular case of a more general theorem.

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