Calculus of Variations and Geometric Measure Theory

A. Bach

Anisotropic free-discontinuity functionals as the $\Gamma$-limit of second-order elliptic functionals

created by bach on 07 Oct 2016
modified on 07 Oct 2020


Published Paper

Inserted: 7 oct 2016
Last Updated: 7 oct 2020

Journal: ESAIM: Control Optim. Calc. Var.
Volume: 24
Year: 2018
Doi: 10.1051/cocv/2017027


We provide an approximation result for free-discontinuity functionals of the form \[\mathcal{F}(u)=\int_\Omega f(x,u,\nabla u)dx+\int_{S_u\cap\Omega}\theta(x,\nu_u)d\mathcal{H}^{n-1},\quad u\in SBV^2(\Omega),\] where $f$ is quadratic in the gradient-variable and $\theta$ is an arbitrary smooth Finsler metric. The approximating functionals are of Ambrosio-Tortorelli type and depend on the Hessian of the edge variable through a suitable nonhomogeneous metric $\phi$.

Keywords: $\Gamma$-convergence, Finsler metrics, Ambrosio-Tortorelli approximation, anisotropic free-discontinuity functionals