Homogenization of a parabolic Dirichlet problem by a method of Dahlberg

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 ${\partial }_t{u}_{\epsilon }(X,t)-\text{div}(A(X/\epsilon )\nabla u_{\epsilon }(X,t))$ in Lipschitz domains with $L^p$-boundary data.

21 pages

References in corpus (2)

• Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients
• Solvability of the Dirichlet, Neumann and the regularity problems for parabolic equations with Hölder continuous coefficients

Cited by in corpus (1)

• A structure theorem for elliptic and parabolic operators with applications to homogenization of operators of Kolmogorov type