I will discuss the homogenization of periodic oscillating Dirichlet
boundary problems in general domains for second order uniformly elliptic
equations. These problems are connected with the study of boundary layers
in fluid mechanics and with the study of higher order asymptotic expansions
in interior homogenization theory. The talk will be aimed at a general
audience. I will explain some recent progress about the continuity
properties of the homogenized problem which displays a sharp contrast
between the case of linear and nonlinear interior equations. This is based
on joint work with Inwon Kim.
The existence and uniqueness of Gevrey regularity solution for a class of nonlinear
bistable gradient flows, with the energy decomposed into purely convex and concave parts,
such as epitaxial thin film growth and square phase field crystal models, are discussed in this talk.
The polynomial pattern of the nonlinear terms in the chemical potential enables one to derive a
local in time solution with Gevrey regularity, with the existence time interval length dependent
on certain functional norms of the initial data. Moreover, a detailed Sobolev estimate for the gradient
equations results in a uniform in time bound, which in turn establishes a global in
time solution with Gevrey regularity. An extension to a system of gradient flows,
such as the three-component Cahn-Hilliard equations, is also addressed in this talk.
Abstract: We discuss some homogenization problems of Hamilton-Jacobi equations in time-dependent (dynamic) random environments, where the coefficients of PDEs are highly oscillatory in the space and time variables. We consider both first order and second order equations, and
emphasize how to overcome the difficulty imposed by the lack of coercivity in the time derivative. In the first order case with linear growing Hamiltonian, periodicity in either the space or the time variable is assumed; in the second order case with at most quadratic growing Hamiltonian, uniform ellipticity of the second order term is assumed.