Homogenization of a parabolic Dirichlet problem by a method of Dahlberg
Alejandro J. Castro, Martin Strömqvist · LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Científicas) · 2016
Consider the linear parabolic operator in divergence form $$\mathcal{H} u =\partial_t u(X,t)-\text{div}(A(X) abla u(X,t)).$$ 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 $\partial_t u_\varepsilon(X,t)-\text{div}(A(X/\varepsilon) abla u_\varepsilon(X,t))$ in Lipschitz domains with $L^p$-boundary data.