Journal Article

Polyconvex functionals and maximum principle

Esposito, Luca, Leonetti, Francesco, Carozza, Menita, Giova, Raffaella

2023

Overview

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.