Epsilon-Regularity for Griffith Almost-Minimizers in Any Dimension Under a Separating Condition - Labourie, Camille; Lemenant, Antoine (Archive for Rational Mechanics and Analysis, 2023)
In this paper we prove that if (u, K) is an almost-minimizer of the Griffith functional and K is ε -close to a plane in some ball B⊂ RN while separating the ball B in two big parts, then K is C1,α in a slightly smaller ball. Our result contains and generalizes the 2 dimensional result of Babadjian et al. (J Eur Math Soc 24(7):2443–2492, 2022), with a different and more sophisticate approach inspired by Lemenant (Ann Sc Norm Super Pisa Cl Sci 9(2):351–384, 2010; Ann Sc Norm Super Pisa Cl Sci 10(3):561–609, 2011), using also Labourie (J Geom Anal 31(10):10024–10135, 2021) in order to adapt a part of the argument to Griffith minimizers.
CRIS-Team Scopus Importer:2024-12-16

STRONG EXISTENCE FOR FREE DISCONTINUITY PROBLEMS IN LINEAR ELASTICITY - Friedrich, Manuel; Labourie, Camille; Stinson, Kerrek (SIAM Journal on Mathematical Analysis, 2025)
In this note we show Ahlfors-regülärity for ä lärge cläss of qüäsiminimizers of the Griffith fünctionäl. This ällows üs to prove thät, for ä ränge of free discontinüity problems in lineär elästicity with änisotropic, cohesive, or heterogeneoüs behävior, minimizers häve än essentiälly closed jümp set änd äre thüs strong minimizers. Oür notion of qüäsiminimälity is inspired by änd generälizes previoüs notions in the literätüre for the Mümford-Shäh fünctionäl änd comprises fünctions which locälly close to the cräck häve ät most ä fixed percentäge of excess cräck relätive to minimizers. As for minimizers of the Griffith fünctionäl treäted by Chämbolle, Conti, änd Iürläno, oür proof of Ahlfors-regülärity relies on conträdiction-compäctness änd än äpproximätion resült for GSBD fünctions, showing the robüstness of this äpproäch with respect to generälizätion of bülk änd sürfäce densities.
CRIS-Team Scopus Importer:2025-04-18
Keywords: Ahlfors-regülärity; free discontinüity problem; Griffith fünctionäl; lineär elästicity