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

We provide an approximation result for free-discontinuity functionals of the form         𝓕(u) = ∫Ωf(x, u, ∇u)dx + ∫Su∩Ωθ(x, νu)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...

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Published inESAIM. Control, optimisation and calculus of variations Vol. 24; no. 3; pp. 1107 - 1139
Main Author Bach, Annika
Format Journal Article
LanguageEnglish
Published Les Ulis EDP Sciences 2018
Subjects
49J45
68U10
74G65
Ambrosio-Tortorelli approximation
anisotropic free-discontinuity functionals
Control theory
Discontinuity
Finsler metrics
Mathematical analysis
Γ-convergence
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Summary:We provide an approximation result for free-discontinuity functionals of the form         𝓕(u) = ∫Ωf(x, u, ∇u)dx + ∫Su∩Ωθ(x, νu)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.
Bibliography:istex:A70910AE7104486B105F40CA3E31F99DE0C096D1
publisher-ID:cocv160139
href:https://www.esaim-cocv.org/articles/cocv/abs/2018/03/cocv160139/cocv160139.html
ark:/67375/80W-7D7ZJRTJ-H
ISSN:1292-8119
1262-3377
DOI:10.1051/cocv/2017027