Anisotropic free-discontinuity functionals as the Γ-limit of second-order elliptic functionals

Angewandte Mathematik, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, 48149 Münster, Germany

^a Corresponding author: a_bach10@uni-muenster.de

Received: 10 October 2016
Accepted: 17 March 2017

Abstract

We provide an approximation result for free-discontinuity functionals of the form

        𝓕(u) = ∫_Ωf(x, u, ∇u)dx + ∫_Su∩Ωθ(x, ν_u)d𝓗ⁿ⁻¹,     u ∈ SBV²(Ω),

where f is quadratic in the gradient-variable and θ 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 ϕ.