Homogenization of a parabolic Dirichlet problem by a method of Dahlberg

TL;DR

This paper proves well-posedness of the Dirichlet problem for a class of parabolic operators with periodic coefficients using Dahlberg's method, enabling homogenization in Lipschitz domains with L^p boundary data.

Contribution

It introduces a Dahlberg-based approach to establish well-posedness and homogenization results for parabolic equations with periodic, Dini-continuous coefficients.

Findings

Well-posedness of Dirichlet problem in upper half-plane for L^p data.

Homogenization of parabolic equations with periodic coefficients in Lipschitz domains.

Extension of Dahlberg's method to parabolic operators with Dini-type conditions.

Abstract

Consider the linear parabolic operator in divergence form $$\mathcal{H} u =\partial_t u(X,t)-\text{div}(A(X)\nabla u(X,t)).$$ We employ a method of Dahlberg to show that the Dirichlet problem for $\mathcal{H}$ in the upper half plane is well-posed for boundary data in $L^p$, for any elliptic matrix of coefficients $A$ which is periodic and satisfies a Dini-type condition. This result allows us to treat a homogenization problem for the equation $\partial_t u_\varepsilon(X,t)-\text{div}(A(X/\varepsilon)\nabla u_\varepsilon(X,t))$ in Lipschitz domains with $L^p$-boundary data.