AI 中文总结
该研究将齐次Dirichlet问题的L^p可解性刻画推广到含混合维边界的区域中一类退化椭圆算子，拓展了可解性理论的适用范围。
AI 中文摘要
我们将齐次Dirichlet问题的L^p可解性的若干刻画推广到定义在R^n中一大类开集Ω上的（可能退化的）椭圆算子L=-div(wA∇)。该框架不仅包含Lipschitz区域中的一致椭圆算子，还涵盖Caffarelli-Sylvestre型算子及非(n-1)维的边界∂Ω等。我们证明，对于1<p<∞，
英文摘要
We extend several characterizations of the $L^p$-solvability of the homogeneous Dirichlet problem to (possibly degenerate) elliptic operators $L=-\textrm{div}(wA\nabla)$ defined on a large class of open sets $Ω$ in $\mathbb{R}^n$. This framework encompasses not only uniformly elliptic operators in Lipschitz domains, but also Caffarelli-Sylvestre-type operators, and boundaries $\partialΩ$ that are not $(n-1)$-dimensional, for instance. We prove that, for $p\in (1,\infty)$, the $L^p$-solvability of the homogeneous Dirichlet problem is equivalent to the solvability of the Poisson-Dirichlet problem $Lu=wf-\textrm{div}(wF)$, \emph{i.e.} to the existence of a solution $u$ satisfying a $L^p$ non-tangential estimate whenever $f$ and $F$ belong to suitable weighted $L^p$ tent spaces. Furthermore, it is also equivalent to the solvability of the Poisson-Regularity problem for data in appropriate weighted $L^{p'}$ tent spaces, where estimates are obtained for $\nabla u$ rather than $u$.
