On the homogenization of second order level set PDE in periodic media

Abstract

This paper analyzes two classes of second order level set PDE in periodic media in the parabolic scaling. First, we study fully nonlinear geometric operators under general assumptions in dimension $d = 2$ and prove that the associated equations homogenize in this case. Next, we treat a class of quasi-linear geometric operators in arbitrary dimensions $d \geq 2$. In this setting, by adapting arguments form the study of oscillating boundary value problems, we prove that the effective coefficients are generically discontinuous in all dimensions $d \geq 3$. This necessitates a study of level set PDE driven by operators that are discontinuous at every rational direction on the sphere. We prove that, in fact, the effective operators so obtained do have a comparison principle and, thus, homogenization occurs. Finally, we investigate the connection between the effective mobility obtained in the quasi-linear case and linear response, drawing a connection between our results and those obtained in the hyperbolic scaling.