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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Abstract 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.
AbstractList We provide an approximation result for free-discontinuity functionals of the form          ( u ) = ∫ Ω f ( x, u,  ∇ u )d x  + ∫ S u ∩ Ω θ ( x, ν u )d 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 .
We provide an approximation result for free-discontinuity functionals of the form ??(u) = ∫Ωf(x, u, ∇u)dx + ∫Su∩Ωθ(x, νu)d??n−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.
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.
Author Bach, Annika
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Cites_doi 10.1137/15M1020848
10.1137/S0036141099356830
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10.1007/BF00052492
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Snippet 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...
We provide an approximation result for free-discontinuity functionals of the form          ( u ) = ∫ Ω f ( x, u,  ∇ u )d x  + ∫ S u ∩ Ω θ ( x, ν u )d n −1...
We provide an approximation result for free-discontinuity functionals of the form ??(u) = ∫Ωf(x, u, ∇u)dx + ∫Su∩Ωθ(x, νu)d??n−1, u ∈ SBV2(Ω), where f is...
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SubjectTerms 49J45
68U10
74G65
Ambrosio-Tortorelli approximation
anisotropic free-discontinuity functionals
Control theory
Discontinuity
Finsler metrics
Mathematical analysis
Γ-convergence
Title Anisotropic free-discontinuity functionals as the Γ-limit of second-order elliptic functionals
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