We consider a family of doubly nonlinear evolution equations that is given by families of convex dissipation potentials, nonconvex energy functionals, and external forces parametrized by a small parameter ε. For each of these problems, we introduce the so-called weighted energy-dissipation (WED) functional, whose minimizers correspond to solutions of an elliptic-in-time regularization of the target problems with regularization parameter δ. We investigate the relation between the Γ-convergence of the WED functionals and evolutionary Γ-convergence of the associated systems. More precisely, we deal with the limits δ → 0, ε → 0, as well as δ + ε → 0 either in the sense of Γ-convergence of functionals or in the sense of evolutionary Γ-convergence of functional-driven evolution problems, or both. Additionally, we provide some quantitative estimates on the rate of convergence for the limit ε → 0, in the case of quadratic dissipation potentials and uniformly λ-convex energy functionals. Finally, we discuss a homogenization and a dimension reduction problem as examples of application.
Accepté le :
DOI : 10.1051/cocv/2018023
Keywords: Doubly nonlinear evolution, weighted-energy-dissipation principle, evolutionary Γ-convergence, variational principle, homogenization, dimension reduction
Liero, Matthias 1 ; Melchionna, Stefano 1
@article{COCV_2019__25__A36_0,
author = {Liero, Matthias and Melchionna, Stefano},
title = {The weighted energy-dissipation principle and evolutionary {\ensuremath{\Gamma}-convergence} for doubly nonlinear problems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2019},
publisher = {EDP Sciences},
volume = {25},
doi = {10.1051/cocv/2018023},
zbl = {1437.58015},
mrnumber = {4003464},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2018023/}
}
TY - JOUR AU - Liero, Matthias AU - Melchionna, Stefano TI - The weighted energy-dissipation principle and evolutionary Γ-convergence for doubly nonlinear problems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2018023/ DO - 10.1051/cocv/2018023 LA - en ID - COCV_2019__25__A36_0 ER -
%0 Journal Article %A Liero, Matthias %A Melchionna, Stefano %T The weighted energy-dissipation principle and evolutionary Γ-convergence for doubly nonlinear problems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2018023/ %R 10.1051/cocv/2018023 %G en %F COCV_2019__25__A36_0
Liero, Matthias; Melchionna, Stefano. The weighted energy-dissipation principle and evolutionary Γ-convergence for doubly nonlinear problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 36. doi: 10.1051/cocv/2018023
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