Boundary value problems in Lipschitz domains for equations with lower order coefficients

TL;DR

This paper investigates boundary value problems for second order elliptic equations with lower order coefficients in Lipschitz domains, establishing existence and uniqueness results using layer potential methods without symmetry or smallness constraints.

Contribution

It provides new existence and uniqueness results for the $R_2$ Regularity and $D_2$ Dirichlet problems for elliptic equations with lower order terms, relaxing symmetry and smallness assumptions.

Findings

Established existence and uniqueness for $R_2$ problem with non-symmetric coefficients.

Proved existence and uniqueness for $D_2$ problem for adjoint equations.

No smallness assumptions on lower order coefficients were needed.

Abstract

We use the method of layer potentials to study the $R_2$ Regularity problem and the $D_2$ Dirichlet problem for second order elliptic equations of the form $\mathcal{L}u=0$, with lower order coefficients, in bounded Lipschitz domains. For $R_2$ we establish existence and uniqueness assuming that $\mathcal{L}$ is of the form $\mathcal{L}u=-\text{div}(A\nabla u+bu)+c\nabla u+du$, where the matrix $A$ is uniformly elliptic and H\"older continuous, $b$ is H\"older continuous, and $c,d$ belong to Lebesgue classes and they satisfy either the condition $d\geq\text{div}b$, or $d\geq\text{div}c$ in the sense of distributions. In particular, $A$ is not assumed to be symmetric, and there is no smallness assumption on the norms of the lower order coefficients. We also show existence and uniqueness for $D_2$ for the adjoint equations $\mathcal{L}^tu=0$.