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We present new abstract results on the interrelation between the minimizing movement scheme for gradient flows along a sequence of Γ-converging functionals and the gradient flow motion for the corresponding limit functional, in a general metric space. We are able to allow a relaxed form of minimization in each step of the scheme, and so we present new relaxation results too.
Fleißner, Florentine 1
@article{COCV_2019__25__A28_0,
author = {Flei{\ss}ner, Florentine},
title = {\ensuremath{\Gamma}-convergence and relaxations for gradient flows in metric spaces: a minimizing movement approach},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {25},
year = {2019},
doi = {10.1051/cocv/2017035},
zbl = {1442.49017},
mrnumber = {3990648},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2017035/}
}
TY - JOUR AU - Fleißner, Florentine TI - Γ-convergence and relaxations for gradient flows in metric spaces: a minimizing movement approach JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2017035/ DO - 10.1051/cocv/2017035 LA - en ID - COCV_2019__25__A28_0 ER -
%0 Journal Article %A Fleißner, Florentine %T Γ-convergence and relaxations for gradient flows in metric spaces: a minimizing movement approach %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2017035/ %R 10.1051/cocv/2017035 %G en %F COCV_2019__25__A28_0
Fleißner, Florentine. Γ-convergence and relaxations for gradient flows in metric spaces: a minimizing movement approach. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 28. doi : 10.1051/cocv/2017035. http://geodesic.mathdoc.fr/articles/10.1051/cocv/2017035/
[1] , and , Curvature-Driven Flows: A Variational Approach. SIAM J. Control Optimiz. 31 (1993) 387–437. | Zbl | MR | DOI
[2] , and , Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lect. Math. ETH Zürich, Birkhäuser (2005). | Zbl | MR
[3] , Variational convergence for functions and operators. Pitman (Advanced Publishing Program), Boston, MA (1984). | Zbl | MR
[4] , Gamma-convergence for Beginners, vol. 22. Oxford University Press (2002). | Zbl | MR | DOI
[5] , Local Minimization, Variational Evolution and Γ-Convergence. Vol. 2094 of Lect. Notes Math. Springer (2012). | Zbl | MR
[6] and , Asymptotic expansions by γ-convergence. Continuum Mech. Therm. 20 (2008) 21–62. | Zbl | MR | DOI
[7] , The finite element method for elliptic problems, In Vol. 4 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam (1978). | Zbl | MR
[8] and , Passing to the limit in maximal slope curves: from a regularized perona–malik equation to the total variation flow. Math. Models Methods Appl. Sci. 22 (2012) 1250017. | Zbl | MR | DOI
[9] An Introduction to Γ-Convergence, vol. 8 of Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser, Boston (1993). | Zbl | MR
[10] and , Lecture notes on gradient flows and optimal transport, Optimal transportation. In vol. 413 of London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, Cambridge (2014) 100–144. | Zbl | MR
[11] New problems on minimizing movements, in Boundary Value Problems for PDE and Applications, edited by and . Masson (1993) 81–98. | Zbl | MR
[12] , and , Problems of evolution in metric spaces and maximal decreasing curve. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 68 (1980) 180–187. | Zbl | MR
[13] and , Sulla convergenza degli integrali dellenergia per operatori ellittici del secondo ordine. Boll. Un. Mat. Ital 8 (1973) 391–411. | Zbl | MR
[14] , and , Evolution equations with lack of convexity. Nonlinear Anal. 9 (1985) 1401–1443. | Zbl | MR | DOI
[15] , On the heat flow on metric measure spaces: existence, uniqueness and stability. Calc. Var. Partial Differ. Equ. 39 (2010) 101–120. | Zbl | MR | DOI
[16] , and , Curves of maximal slope and parabolic variational inequalities on nonconvex constraints. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 16 (1989) 281–330. | Zbl | MR | mathdoc-id
[17] , On evolutionary Γ-convergence for gradient systems. Lect. Notes Appl. Math. Mech. Springer (2016). | MR | DOI
[18] , and , Variational convergence of gradient flows and rate-independent evolutions in metric spaces. Milan J. Math. 80 (2012) 381–410. | Zbl | MR | DOI
[19] , and , Γ-limits and relaxations for rate-independent evolutionary problems. Calcul. Variat. Partial Differ. Equ. 31 (2008) 387–416. | Zbl | MR | DOI
[20] , Two variational techniques for the approximation of curves of maximal slope. Tech. Report NA-05/10, Oxford Comp. Lab. Report. Available at: http://web2.comlab.ox.ac.uk/oucl/publications/natr/na-05-10.html (2015).
[21] and , Gradient flows of non convex functionals in Hilbert spaces and applications. ESAIM: COCV 12 (2006) 564–614. | Zbl | MR | mathdoc-id
[22] and , Gamma-convergence of gradient flows with applications to Ginzburg-Landau. Commun. Pure Appl. Math. 57 (2004) 1627–1672. | Zbl | MR | DOI
[23] , Gamma-convergence of gradient flows on hilbert and metric spaces and applications. Discrete Contin. Dyn. Syst. 31 (2011) 1427–1451. | Zbl | MR | DOI
[24] , Sul limite delle soluzioni di problemi di Cauchy relativi all’equazione del calore. Ann. Scuola Norm. Sup. Pisa 21 (1967) 657–699. | Zbl | MR | mathdoc-id
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