L2 Solvability of boundary value problems for divergence form parabolic equations with complex coefficients

Abstract

We consider parabolic operators of the form∂t+L,L=−divA(X,t)∇, in R+n+2:={(X,t)=(x,xn+1,t)∈Rn×R×R:xn+1>0}, n≥1. We assume that A is a (n+1)×(n+1)-dimensional matrix which is bounded, measurable, uniformly elliptic and complex, and we assume, in addition, that the entries of A are independent of the spatial coordinate xn+1 as well as of the time coordinate t. For such operators we prove that the boundedness and invertibility of the corresponding layer potential operators are stable on L2(Rn+1,C)=L2(∂R+n+2,C) under complex, L∞ perturbations of the coefficient matrix. Subsequently, using this general result, we establish solvability of the Dirichlet, Neumann and Regularity problems for ∂t+L, by way of layer potentials and with data in L2, assuming that the coefficient matrix is a small complex perturbation of either a constant matrix or of a real and symmetric matrix.