The regularity problem with a weaker condition on only the transversal direction

Martin Ulmer

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arXiv · 2503.20453

The regularity problem with a weaker condition on only the transversal direction

Martin Ulmer

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Abstract

We study an elliptic operator $L:=\mathrm{div}(A\nabla \cdot)$ on the upper half space. It is known that if the matrix $A$ is independent in the transversal $t$-direction, then the regularity boundary value problem is solvable with data in a Sobolev space. In the present paper we improve on the $t$-independence condition by introducing a mixed $L^1-L^\infty$ condition that only depends on $\partial_t A$, the derivative of $A$ in transversal direction. This condition is different from other conditions in the literature and has already been proven to imply solvability of the Dirichlet boundary value problem.

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Martin Ulmer. 2025-03-26. The regularity problem with a weaker condition on only the transversal direction. https://arxiv.org/abs/2503.20453

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