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We provide an approximation result for free-discontinuity functionals of the form 𝓕( u ) = ∫ Ω f ( x, u, ∇ u )d x + ∫ S u ∩ Ω θ ( x, ν u )d𝓗 n −1 , u ∈ SBV 2 ( 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 .
Annika Bach (Mon,) studied this question.