$L^2$ 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 ${\mathbb{R}}_{+}^{n+2}:=\{(X,t)=(x,x_{n+1},t)\in {\mathbb{R}}^n\times \mathbb{R}\times \mathbb{R}:x_{n+1}>0\}$, $n\ge 1$. 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.

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