# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2022055
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## Stability of a class of action functionals depending on convex functions

 Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy

*Corresponding author: Camillo Brena

Received  June 2021 Revised  March 2022 Early access April 2022

Fund Project: Work supported by the PRIN 2017 project "Gradient flows, Optimal Transport and Metric Measure Structures"

We study the stability of a class of action functionals induced by gradients of convex functions with respect to Mosco convergence, under mild assumptions on the underlying space.

Citation: Luigi Ambrosio, Camillo Brena. Stability of a class of action functionals depending on convex functions. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022055
##### References:
 [1] L. Ambrosio, A. Baradat and Y. Brenier, $\Gamma$-convergence for a class of action functionals induced by gradients of convex functions, Rend. Lincei. Mat. Appl., 32 (2021), 97-108.  doi: 10.4171/RLM/928. [2] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2$^{nd}$ edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. [3] B. Bač ák, Convex Analysis and Optimization in Hadamard Spaces, De Gruyter Series in Nonlinear Analysis and Applications, Berlin, 2014. doi: 10.1515/9783110361629. [4] M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-12494-9. [5] G. Clerc, G. Conforti and I. Gentil, On the variational interpretation of local logarithmic Sobolev inequalities, preprint, 2020, arXiv: 2011.05207. [6] S. Daneri and G. Savaré, Lecture notes on gradient flows and optimal transport, In London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, 413 (2014), 100–144. [7] U. Mayer, Gradient flows on nonpositively curved metric spaces and harmonic maps, Comm. Anal. Geom., 6 (1998), 199-253.  doi: 10.4310/CAG.1998.v6.n2.a1. [8] L. Monsaingeon, L. Tamanini and D. Vorotnikov, The dynamical Schrödinger problem in abstract metric spaces, preprint, 2020, arXiv: 2012.12005. [9] M. Muratori and G. Savaré, Gradient flows and evolution variational inequalities in metric spaces. Ⅰ: Structural properties, J. Funct. Anal., 278 (2020), 108347, 67 pp. doi: 10.1016/j.jfa.2019.108347.

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##### References:
 [1] L. Ambrosio, A. Baradat and Y. Brenier, $\Gamma$-convergence for a class of action functionals induced by gradients of convex functions, Rend. Lincei. Mat. Appl., 32 (2021), 97-108.  doi: 10.4171/RLM/928. [2] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2$^{nd}$ edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. [3] B. Bač ák, Convex Analysis and Optimization in Hadamard Spaces, De Gruyter Series in Nonlinear Analysis and Applications, Berlin, 2014. doi: 10.1515/9783110361629. [4] M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-12494-9. [5] G. Clerc, G. Conforti and I. Gentil, On the variational interpretation of local logarithmic Sobolev inequalities, preprint, 2020, arXiv: 2011.05207. [6] S. Daneri and G. Savaré, Lecture notes on gradient flows and optimal transport, In London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, 413 (2014), 100–144. [7] U. Mayer, Gradient flows on nonpositively curved metric spaces and harmonic maps, Comm. Anal. Geom., 6 (1998), 199-253.  doi: 10.4310/CAG.1998.v6.n2.a1. [8] L. Monsaingeon, L. Tamanini and D. Vorotnikov, The dynamical Schrödinger problem in abstract metric spaces, preprint, 2020, arXiv: 2012.12005. [9] M. Muratori and G. Savaré, Gradient flows and evolution variational inequalities in metric spaces. Ⅰ: Structural properties, J. Funct. Anal., 278 (2020), 108347, 67 pp. doi: 10.1016/j.jfa.2019.108347.
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