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

TL;DR

This paper proves the stability of layer potential operators for complex parabolic equations with certain coefficient conditions and establishes boundary value problem solvability in L^2 spaces under small complex perturbations.

Contribution

It demonstrates the stability of layer potential operators for complex parabolic operators with specific independence conditions and proves solvability of boundary value problems under small complex perturbations.

Findings

Layer potential operators are stable under complex, L-infinity perturbations.

Solvability of Dirichlet, Neumann, and Regularity problems is established for small complex perturbations.

Results extend to operators with coefficients close to constant or symmetric real matrices.

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.