A new solvability condition for $L^p$ boundary value problems

Abstract: We are discussing the elliptic operator $L:=\mathrm{div}(A\nabla\cdot)$ and wonder which types of matrices $A$ yield solvability of $L^p$ boundary value problems. It is well-known that the DKP or Carleson condition implies solvability for the Dirichlet and the regularity boundary value problem. Equally, if the domain is the upper half space, independence of the transversal direction $t$ gives solvability of these boundary value problems.
On the upper half space $\mathbb{R}^{n+1}_+$, we would like to introduce a different sufficient condition for solvability which is given by a mixed $L^1-L^\infty$ condition. This condition generalizes the class of $t$-independent operators and also implies solvability of the $L^p$ Dirichlet and regularity boundary value problem. If we are in the setting of the upper half plane $\mathbb{R}^{2}_+$, we even obtain a stronger result for the Dirichlet problem by the same proof strategy: If $A$ satisfies an $L^1$-Carleson condition on $\partial_t A$, then we obtain $\omega\in A_\infty(\sigma)$ or solvability of the $L^p$ Dirichlet problem.