Threshold phenomenon for homogenized fronts in random elastic media

Patrick W. Dondl; Martin Jesenko

doi:10.3934/dcdss.2020329

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2021, Volume 14, Issue 1: 353-372. Doi: 10.3934/dcdss.2020329

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Threshold phenomenon for homogenized fronts in random elastic media

Patrick W. Dondl^1, , and
Martin Jesenko^1,

1.
Abteilung für Angewandte Mathematik, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Str. 10, 79104 Freiburg, Germany

^* Corresponding author: Patrick W. Dondl
^* Corresponding author: Patrick W. Dondl

Dedicated to Alexander Mielke on the occasion of his 60th birthday

Received: April 30, 2019

Revised: October 31, 2019

Early access: April 2020

Published: January 2021

This work is supported by EPSRC grant EP/M028682/1 'Effective properties of interface evolution in a random environment.'

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Abstract

We consider a model for the motion of a phase interface in an elastic medium, for example, a twin boundary in martensite. The model is given by a semilinear parabolic equation with a fractional Laplacian as regularizing operator, stemming from the interaction of the front with its elastic environment. We show that the presence of randomly distributed, localized obstacles leads to a threshold phenomenon, i.e., stationary solutions exist up to a positive, critical driving force leading to a stick-slip behaviour of the phase boundary. The main result is proved by an explicit construction of a stationary viscosity supersolution to the evolution equation and is based on a percolation result for the obstacle sites. Furthermore, we derive a homogenization result for such fronts in the case of the half-Laplacian in the pinning regime.

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Mathematics Subject Classification: 35D40, 35R11, 74A40, 74N20.

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Figure 1. Decomposition of the base space for $ n = 2 $

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Figure 2. Cross-sections of the decomposition of $ {\mathbb R}^{n+1} $ above the horizontal hyperplane (left) and slightly perturbed one (right)

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