Solvability of elliptic systems with square integrable boundary data

TL;DR

This paper investigates the conditions under which second order elliptic systems with complex coefficients are well posed for boundary value problems with square integrable data, establishing openness and specific cases of solvability.

Contribution

It proves that the set of coefficients for which the boundary value problems are well posed is open, and identifies classes of coefficients (Hermitean, block, constant) where solvability is guaranteed.

Findings

The set of coefficients with well-posed boundary problems is open.

Boundary value problems are well posed for Hermitean, block, or constant coefficients.

Methods extend to more general PDE systems and perturbation results for differential forms.

Abstract

We consider second order elliptic divergence form systems with complex measurable coefficients $A$ that are independent of the transversal coordinate, and prove that the set of $A$ for which the boundary value problem with $L_2$ Dirichlet or Neumann data is well posed, is an open set. Furthermore we prove that these boundary value problems are well posed when $A$ is either Hermitean, block or constant. Our methods apply to more general systems of PDEs and as an example we prove perturbation results for boundary value problems for differential forms.