The Regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients

Abstract

The present paper establishes a certain duality between the Dirichlet and Regularity problems for elliptic operators with $t$-independent complex bounded measurable coefficients ($t$ being the transversal direction to the boundary). To be precise, we show that the Dirichlet boundary value problem is solvable in $L^{p^{\prime }}$, subject to the square function and non-tangential maximal function estimates, if and only if the corresponding Regularity problem is solvable in $L^p$. Moreover, the solutions admit layer potential representations. In particular, we prove that for any elliptic operator with $t$-independent real (possibly non-symmetric) coefficients there exists a $p>1$ such that the Regularity problem is well-posed in $L^p$.

References in corpus (3)

• Analyticity of layer potentials and $L^2$ solvability of boundary value problems for divergence form elliptic equations with complex $L^{\infty }$ coefficients
• Elliptic Partial Differential Equations with Complex Coefficients
• The regularity problem for elliptic operators with boundary data in Hardy-Sobolev space $H{S}^1$