We provide an approximation result for free-discontinuity functionals of the form \n \mathcal{F}\left(u\right)={\int}_{\Omega}f(x,u,\phantom{\rule{0.166667em}{0ex}}\nabla u)\mathrm{d}x+{\int}_{{s}_{u}\cap \Omega}\theta (x,{\nu}_{u})\mathrm{d}{\mathscr{H}}^{n-1},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}u\in SB{V}^{2}\left(\Omega \right)\n
where \n f\n is quadratic in the gradient-variable and \n \theta \n 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 \n \varphi \n.
Keywords: Γ-convergence, Ambrosio-Tortorelli approximation, anisotropic free-discontinuity functionals, Finsler metrics
@article{COCV_2018__24_3_1107_0, author = {Bach, Annika}, title = {Anisotropic free-discontinuity functionals as the {\ensuremath{\Gamma}-limit} of second-order elliptic functionals}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1107--1139}, publisher = {EDP-Sciences}, volume = {24}, number = {3}, year = {2018}, doi = {10.1051/cocv/2017027}, zbl = {1412.49032}, mrnumber = {3877195}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2017027/} }
TY - JOUR AU - Bach, Annika TI - Anisotropic free-discontinuity functionals as the Γ-limit of second-order elliptic functionals JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1107 EP - 1139 VL - 24 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2017027/ DO - 10.1051/cocv/2017027 LA - en ID - COCV_2018__24_3_1107_0 ER -
%0 Journal Article %A Bach, Annika %T Anisotropic free-discontinuity functionals as the Γ-limit of second-order elliptic functionals %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1107-1139 %V 24 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2017027/ %R 10.1051/cocv/2017027 %G en %F COCV_2018__24_3_1107_0
Bach, Annika. Anisotropic free-discontinuity functionals as the Γ-limit of second-order elliptic functionals. ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 3, pp. 1107-1139. doi : 10.1051/cocv/2017027. http://archive.numdam.org/articles/10.1051/cocv/2017027/
[1] Sobolev Spaces. Academic Press, New York (1975) | MR | Zbl
,[2] A compactness theorem for a new class of functions of bounded variation. Boll. Un. Mat. Ital. 3 (1989a) 857–881 | MR | Zbl
,[3] Variational problems in SBV and image segmentation. Acta Appl. Math. 17 (1989b) 1–40 | DOI | MR | Zbl
,[4] Existence theory for a new class of variational problems. Arch. Ration. Mech. Anal. 111 (1990) 291–322 | DOI | MR | Zbl
,[5] Functions of Bounded Variation and Free Discontinuity Problems. Oxford Math. Monogr. Clarendon Press, New York (2000) | DOI | MR | Zbl
, and ,[6] Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Commun. Pure Appl. Math. 43 (1990) 999–1036 | DOI | MR | Zbl
and ,[7] On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. 6 (1992) 105–123 | MR | Zbl
and ,[8] Coupled second order singular perturbations for phase transitions. Nonlinearity 26 (2013) 1271–1311 | DOI | MR | Zbl
, , and ,[9] Elliptic approximations of prescribed mean curvature surfaces in Finsler geometry. Asymptotic Anal. 22 (2000) 87–111 | MR | Zbl
and ,[10] Anisotropic motion by mean curvature in the context of Finsler geometry. Hokkaido Math. J. 25 (1996) 537–566 | DOI | MR | Zbl
and ,[11] Some results on surface measures in calculus of variations. Ann. Mat. Pura Appl. 170 (1996) 329–359 | DOI | MR | Zbl
, and ,[12] A global method for relaxation in W^{1,p} and in SBV _{p}. Arch. Ration. Mech. Anal. 165 (2002) 187–242 | DOI | MR | Zbl
, , and ,[13] Approximation of Free-discontinuity Problems. Lecture Notes in Mathematics. Springer Verlag, Berlin (1998) | DOI | MR | Zbl
,[14] Γ-convergence for Beginners. Vol. 22 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (2002) | MR | Zbl
,[15] Second-Order Edge-Penalization in the Ambrosio-TortorelliFunctional. SIAM Multiscale Model. Simul. 13 (2015) 1354–1389 | DOI | MR | Zbl
, and ,[16] Singular perturbation models in phase transitions for second-order materials. Indiana Univ. Math. J. 60 (2011) 591–639 | DOI | MR | Zbl
, , and ,[17] Asymptotic analysis of a second-order singular perturbation model for phase transitions Calc. Var. 41 (2011) 127–150 | DOI | MR | Zbl
, and ,[18] A density result in SBV with respect to non-isotropic energies. Nonlinear Anal. 38 (1999) 585–604 | DOI | MR | Zbl
and ,[19] T. Esposito, Second-order approximation of free-discontinuity problems with linear growth. available online from http://cvgmt.sns.it/media/doc/paper/3162/E2016.pdf. | MR
[20] On the variational approximation of free-discontinuity problems in the vectorial case. Math. Models Methods App. Sci. 11 (2001) 663–684 | DOI | MR | Zbl
,[21] Second order singular perturbation models for phase transitions. SIAM J. Math. Anal. 31 (2000) 1121–1143 | DOI | MR | Zbl
and ,[22] Quasi-convex integrands and lower semicontinuity in L^{1}. SIAM J. Math. Anal. 23 (1992) 1081–1098 | DOI | MR | Zbl
and ,[23] Un nuovo funzionale del calcolo delle variazioni. Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 82 (1988) 199–210 | Zbl
and ,[24] Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991) | DOI | MR
and ,[25] Lower semicontinuity in spaces of weakly differentiable functions. Math. Ann. 313 (1999) 653–710 | DOI | MR | Zbl
,[26] The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98 (1987) 123–142 | DOI | MR | Zbl
,[27] Un esempio di Γ-convergenza. Bol. Unione. Mat. Ital. 14 (1977) 285–299 | MR | Zbl
and ,[28] Optimal approximation by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42 (1989) 577–685 | DOI | MR | Zbl
and ,[29] Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series. Princeton University Press, New Jersey (1970) | MR | Zbl
,[30] Mumford-Shah model for detection of thin structures in an image. Theses, Université d’Orléans, September 2015. URL https://hal.archives-ouvertes.fr/tel-01231219
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