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Homogenization of Cahn–Hilliard-type equations via evolutionary \(\varvec{\Gamma }\)-convergence

  • Matthias Liero
  • Sina ReicheltEmail author
Article

Abstract

In these notes we discuss two approaches to evolutionary \(\Gamma \)-convergence of gradient systems in Hilbert spaces. The formulation of the gradient system is based on two functionals, namely the energy functional and the dissipation potential, which allows us to employ \(\Gamma \)-convergence methods. In the first approach we consider families of uniformly convex energy functionals such that the limit passage of the time-dependent problems can be based on the theory of evolutionary variational inequalities as developed by Daneri and Savaré 2010. The second approach uses the equivalent formulation of the gradient system via the energy-dissipation principle and follows the ideas of Sandier and Serfaty 2004.

We apply both approaches to rigorously derive homogenization limits for Cahn–Hilliard-type equations. Using the method of weak and strong two-scale convergence via periodic unfolding, we show that the energy and dissipation functionals \(\Gamma \)-converge. In conclusion, we will give specific examples for the applicability of each of the two approaches.

Keywords

Evolutionary \(\Gamma \)-convergence Gradient systems Homogenization Cahn–Hilliard equation Evolutionary variational inequality Energy-dissipation principle Two-scale convergence 

Mathematics Subject Classification

35B27 35K55 35K30 35B30 49J40 49J45 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany

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