Stochastic homogenization of $ \Lambda $-convex gradient flows

Martin Heida; Stefan Neukamm; Mario Varga

doi:10.3934/dcdss.2020328

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2021, Volume 14, Issue 1: 427-453. Doi: 10.3934/dcdss.2020328

This issue Previous Article Effective diffusion in thin structures via generalized gradient systems and EDP-convergence Next Article

Stochastic homogenization of $ \Lambda $-convex gradient flows

1.
Fakultät für Mathematik, Technische Universität München, Garching, Germany
2.
Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany
3.
Faculty of Mathematics, Technische Universität Dresden, Dresden, Germany

^* Corresponding author: Mario Varga
^* Corresponding author: Mario Varga

This paper is dedicated to Alexander Mielke on the occasion of his 60th birthday

Received: May 2019

Revised: October 2019

Early access: April 2020

Published: January 2021

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Abstract

In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a $ \Lambda $-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen-Cahn type equations and evolutionary equations driven by the $ p $-Laplace operator with $ p\in (1, \infty) $. The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of ($ \Lambda $-)convex functionals.

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Mathematics Subject Classification: Primary: 49J40, 74Q10; Secondary: 35K57.

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