$L^2$ Solvability of boundary value problems for divergence form parabolic equations with complex coefficients

Abstract

We consider parabolic operators of the form $$\partial_t+\mathcal{L},\ \mathcal{L}=-\mbox{div}\, A(X,t)\nabla,$$ in $\mathbb R_+^{n+2}:=\{(X,t)=(x,x_{n+1},t)\in \mathbb R^{n}\times \mathbb R\times \mathbb R:\ x_{n+1}>0\}$, $n\geq 1$. We assume that $A$ is a $(n+1)\times (n+1)$-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 $x_{n+1}$ 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 $L^2(\mathbb R^{n+1},\mathbb C)=L^2(\partial\mathbb R^{n+2}_+,\mathbb C)$ under complex, $L^\infty$ perturbations of the coefficient matrix. Subsequently, using this general result, we establish solvability of the Dirichlet, Neumann and Regularity problems for $\partial_t+\mathcal{L}$, by way of layer potentials and with data in $L^2$, assuming that the coefficient matrix is a small complex perturbation of either a constant matrix or of a real and symmetric matrix.