Publikation: Partial regularity for symmetric quasiconvex functionals on BD
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Zusammenfassung
We establish the first partial regularity results for (strongly) symmetric quasiconvex functionals of linear growth on BD, the space of functions of bounded deformation. By Rindler's foundational work [64], symmetric quasiconvexity is the pivotal notion as regards sequential weak*-lower semicontinuity and hence for the existence of minima of the relaxed functionals on BD. The overarching main difficulty here is the lack of Korn's Inequality in the L1-setting, hereby implying that the BD-case is genuinely different from the study of variational integrals on BV. Unlike for superlinear growth, symmetric quasiconvex functionals, where we establish partial regularity by direct reduction to the full gradient case by Korn-type inequalities, such a reduction does not work in the linear growth case and identifies the latter as the only situation requiring a treatment on its own.
Zusammenfassung in einer weiteren Sprache
Nous établissons les premiers résultats de régularité partielles pour des fonctionnelles quasi-convexes (fortement) symétriques ayant une croissance linéaire sur BD, l'espace des fonctions dont la déformation est bornée. De part les travaux précurseurs de Rindler [64], la quasi-convexité symétrique est la notion centrale relativement à la semi-continuité inférieure faible-* et donc pour l'existence des minima de fonctionnelles relaxées sur BD. La difficulté générale ici est due à l'absence de l'inégalité de Korn dans le cadre fonctionel L1 , qui de ce fait révèle une différence fondamentale entre le cas BD et celui relatif à l'étude des intégrales variationnelles sur BV. Contrairement aux fonctionnelles quasi-convexes symétriques à croissance superlinéaire pour lesquelles nous établissons la régularité partielle via réduction directe au cas du gradient complet grâce aux inégalités de type Korn, une telle réduction ne peut être implémentée dans le cas de la croissance linéaire et identifie ce dernier comme étant la seule situation qui nécéssite un traitement particulier.
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GMEINEDER, Franz, 2021. Partial regularity for symmetric quasiconvex functionals on BD. In: Journal de Mathématiques Pures et Appliquées. Elsevier. 2021, 145, pp. 83-129. ISSN 0021-7824. eISSN 1776-3371. Available under: doi: 10.1016/j.matpur.2020.09.005BibTex
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doi={10.1016/j.matpur.2020.09.005},
title={Partial regularity for symmetric quasiconvex functionals on BD},
volume={145},
issn={0021-7824},
journal={Journal de Mathématiques Pures et Appliquées},
pages={83--129},
author={Gmeineder, Franz}
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