Stochastic homogenization of quasilinear PDEs with a spatial degeneracy

Abstract

We investigate stochastic homogenization for some degenerate quasilinear parabolic PDEs. The underlying nonlinear operator degenerates along the space variable, uniformly in the nonlinear term: the degeneracy points correspond to the degeneracy points of a reference diffusion operator on the random medium.

Assuming that this reference diffusion operator is ergodic, we can prove the homogenization property for the quasilinear PDEs, by means of the first-order approximation method. The (nonlinear) limit operator need not be nondegenerate. Concrete examples are provided.