$L^2$ well-posedness of boundary value problems for parabolic systems with measurable coefficients

TL;DR

This paper establishes the well-posedness of boundary value problems for second order parabolic systems with measurable coefficients, introducing a first order approach and solving the Kato square root problem in this context.

Contribution

It develops a novel first order strategy using a parabolic Dirac operator to handle measurable coefficients and non-local derivatives, extending previous results to more general cases.

Findings

Proves well-posedness of boundary value problems in $L^2$-Sobolev spaces.

Introduces a first order approach with a parabolic Dirac operator.

Solves the Kato square root problem for parabolic operators.

Abstract

We prove the first positive results concerning boundary value problems in the upper half-space of second order parabolic systems only assuming measurability and some transversal regularity in the coefficients of the elliptic part. To do so, we introduce and develop a first order strategy by means of a parabolic Dirac operator at the boundary to obtain, in particular, Green's representation for solutions in natural classes involving square functions and non-tangential maximal functions, well-posedness results with data in $L^2$-Sobolev spaces together with invertibility of layer potentials, and perturbation results. In the way, we solve the Kato square root problem for parabolic operators with coefficients of the elliptic part depending measurably on all variables. The major new challenge, compared to the earlier results by one of us under time and transversally independence of the coefficients, is to handle non-local half-order derivatives in time which are unavoidable in our situation.