Boundary value problems for second order elliptic operators with complex coefficients

TL;DR

This paper advances the theory of second order elliptic operators with complex coefficients by establishing solvability of the Regularity boundary value problem and extending the range of $L^p$ solvability for the Dirichlet problem under $p$-ellipticity conditions.

Contribution

It proves the solvability of the Regularity problem for $p$-elliptic operators and extends $L^p$ solvability results for the Dirichlet problem using Shen's theorem.

Findings

Regularity problem solvability established for $p$-elliptic operators.

Extended the $L^p$ range for Dirichlet problem solvability.

Solutions exhibit higher integrability and specific Sobolev space properties.

Abstract

The theory of second order complex coefficient operators of the form $\mathcal{L}=\mbox{div} A(x)\nabla$ has recently been developed under the assumption of $p$-ellipticity. In particular, if the matrix $A$ is $p$-elliptic, the solutions $u$ to $\mathcal{L}u = 0$ will satisfy a higher integrability, even though they may not be continuous in the interior. Moreover, these solutions have the property that $|u|^{p/2-1}u \in W^{1,2}_{loc}$. These properties of solutions were used by Dindo\v{s}-Pipher to solve the $L^p$ Dirichlet problem for $p$-elliptic operators whose coefficients satisfy a further regularity condition, a Carleson measure condition that has often appeared in the literature in the study of real, elliptic divergence form operators. This paper contains two main results. First, we establish solvability of the Regularity boundary value problem for this class of operators, in the same range as that of the Dirichlet problem. The Regularity problem, even in the real elliptic setting, is more delicate than the Dirichlet problem because it requires estimates on derivatives of solutions. Second, the Regularity results allow us to extend the previously established range of $L^p$ solvability of the Dirichlet problem using a theorem due to Z. Shen for general bounded sublinear operators.