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We consider a moving interface that is coupled to an elliptic equation in aheterogeneous medium. The problem is motivated by the study of displacivesolid-solid phase transformations. We show that a nearly flat interface isgiven by the graph of the function $g$ which evolves according to the equation$g_t (x) = -(-\Delta)^{1/2}g (x) + \varphi(x, g(x)) + F$. This equation alsoarises in the study of dislocations and fracture. We show in the periodicsetting that such interfaces exhibit a stick-slip behavior associated withpinning and depinning. Further, we present some numerical evidence that theeffective velocity of the phase boundary scales as the square-root of theexcess macroscopic force above the depinning transition.

Effective behavior of an interface propagating through a periodic elastic medium