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.

中文翻译：

---

随机弹性介质中均质前沿的阈值现象

我们考虑弹性介质（例如，马氏体中的孪晶边界）中相界面运动的模型。该模型由半线性抛物线方程给出，分数阶拉普拉斯算子是正则化运算符，其源于锋面与其弹性环境的相互作用。我们表明，随机分布的局部障碍物的存在会导致阈值现象，即静态解决方案存在高达正临界驱动力，从而导致相边界的粘滑行为。主要结果由对演化方程的平稳粘度超解的显式构造证明，并基于障碍物的渗流结果。此外，在钉扎制度下的半拉普拉斯算例中，我们得出了此类锋面的均质化结果。