# American Institute of Mathematical Sciences

January  2021, 14(1): 353-372. doi: 10.3934/dcdss.2020329

## Threshold phenomenon for homogenized fronts in random elastic media

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

* Corresponding author: Patrick W. Dondl

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

Received  May 2019 Revised  November 2019 Published  April 2020

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

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.

Citation: Patrick W. Dondl, Martin Jesenko. Threshold phenomenon for homogenized fronts in random elastic media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 353-372. doi: 10.3934/dcdss.2020329
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##### References:
Decomposition of the base space for $n = 2$
Cross-sections of the decomposition of ${\mathbb R}^{n+1}$ above the horizontal hyperplane (left) and slightly perturbed one (right)
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