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Free-discontinuity problems: calibration and approximation of solutions
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Nonlocal Approximation of Nonisotropic Free-Discontinuity Problems
SIAM Journal on Applied Mathematics, 1999A large class of problems in fracture mechanics, image segmentation, liquid cristals theory can be formulated as ``free-discontinuity problems'' in which one looks for the minimum of integral functionals of the form: \[ \int_{\Omega}g(x,\nabla u) dx + \int_{S_u}\varphi(x,[u],\nu_u) d{\mathcal{H}}^{n-1}, \] where \(\Omega\) is an open and bounded set in
CORTESANI G, TOADER, Rodica
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Homogenization of free discontinuity problems
Archive for Rational Mechanics and Analysis, 1996Following Griffith's theory, hyperelastic brittle media subject to fracture can be modeled by the introduction, in addition to the elastic volume energy, of a surface term which accounts for crack initiation. In its simplest formulation, the energy of a deformation \(u\) is of the form \[ E(u, K)=\int_{\Omega\setminus K}f(\nabla u)dx+ \lambda{\mathcal ...
Braides A., Defranceschi A., Vitali E.
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REGULAR APPROXIMATION OF FREE-DISCONTINUITY PROBLEMS
Mathematical Models and Methods in Applied Sciences, 2000We consider a class of smooth local nonconvex functionals defined on W2,2(Ω), depending on a small parameter ε and we prove that they converge, as ε tends to 0, to a functional F(u,Ω) with a bulk density depending on the gradient of u and a surface energy concentrated on the jump set of u.
Bouchitté, G., Dubs, C., Seppecher, P.
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Free-discontinuity problems generated by singular perturbation
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1998We show that some free discontinuity problems can be obtained as a limit of nonconvex local functionals with a singular perturbation of higher order.
ALICANDRO, Roberto +2 more
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Monotonicity Formula and Regularity for General Free Discontinuity Problems
Archive for Rational Mechanics and Analysis, 2013zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Bucur, Dorin, Stephan, Luckhaus
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Smooth and Broken Minimizers of Some Free Discontinuity Problems
2017We show that minimizers of free discontinuity problems with energy dependent on jump integrals and Dirichlet boundary conditions are smooth provided a smallness condition is imposed on data. We examine in detail two examples: the elastic-plastic beam and the elastic-plastic plate with free yield lines.
Percivale, Danilo, Tomarelli, Franco
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Sequences of Singularly Perturbed Functionals Generating Free-Discontinuity Problems
SIAM Journal on Mathematical Analysis, 2003This is a very interesting paper. From the Introduction: ``Many models in the fields of fracture mechanics and computer vision lead to free-discontinuity problems, that is, to the minimization of functionals defined in spaces of discontinuous functions (namely, BV and SBV) involving energies with a bulk part and a surface part concentrated along the ...
M. Morini
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Sequences of Non-Local Functionals Which Approximate Free-Discontinuity Problems
Archive for Rational Mechanics and Analysis, 1998The aim of the paper is to develop a general method for the approximation, in the sense of \(\Gamma\)-convergence, of free discontinuity problems \[ \min\biggl\{ \int_\Omega g(x,\nabla u) dx+\int_{S_u\cap\Omega} \varphi(x,[u],\nu_u) d{\mathcal H}^{n-1}: u\in \text{SBV}(\Omega)\biggr\} \] (here \(S_u\) is the approximate discontinuity set, \([u]\) is ...
G. Cortesani
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