THRESHOLD PHENOMENON FOR HOMOGENIZED FRONTS IN RANDOM ELASTIC MEDIA

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.

Automated summary

We investigate a model for the motion of a phase interface in an elastic medium, described by a semilinear parabolic equation with a fractional Laplacian. The presence of randomly distributed, localized obstacles leads to a threshold phenomenon, and a percolation result for the obstacle sites is crucial in proving the main result. Additionally, a homogenization result for such fronts in the pinning regime is derived.