Calculus of Variations and Geometric Measure Theory
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A. Bach

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

created by bach on 07 Oct 2016
modified on 19 Apr 2017

[BibTeX]

Accepted Paper

Inserted: 7 oct 2016
Last Updated: 19 apr 2017

Journal: ESAIM: Control Optim. Calc. Var.
Year: 2017
Doi: 10.1051/cocv/2017027

Abstract:

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


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