The Dirichlet problem as the boundary of the Poisson problem: A sharp approximation result
Mihalis Mourgoglou, Bruno Poggi
TL;DR
The paper addresses the boundary-regularity interplay for elliptic equations on corkscrew domains with $n$-Ahlfors regular boundaries. It first identifies the dual of the Kenig-Pipher-inspired space $N_{2,p}$ as a direct sum of a Carleson-type space and a boundary $L^{p'}$ space, clarifying the boundary data decomposition. Building on this duality and recent Dirichlet/Poisson solvability results, it proves a sharp approximation theorem: for any $g\in L^p(\partial\Omega)$, the Dirichlet solution $u$ with data $g$ can be obtained as the weak-$*$ boundary limit of Poisson solutions $w_j$ with inhomogeneous data $F_j$ in ${\bf C}_{2,p}$, provided the Dirichlet problem is solvable. The approximating data $F_j$ are constructed explicitly via Varopoulos extensions, and the convergence is global (weak-$*$ in $N_{2,p}$) and sharp, not generally improving to weak convergence in $N_{2,p}$. These results connect the boundary and interior problems in a precise, quantitatively sharp framework, and they hold even for non-symmetric, real coefficient matrices and in the unit ball, thereby extending classical Laplacian theory.
Abstract
On a bounded domain $Ω\subset\mathbb R^{n+1}$, $n\geq2$, satisfying the corkscrew condition and with Ahlfors regular boundary, we characterize the dual space to the space ${\bf N}_{2,p}$ of functions $u$ whose Kenig-Pipher modified non-tangential maximal operator $\mathcal N_2(u)$ lies in $L^p(\partialΩ)$, $p\in(1,\infty)$. We find that \[ ({\bf N}_{2,p})^*={\bf C}_{2,p'}\oplus L^{p'}(\partialΩ),\qquad\text{and that}\qquad L^{p'}(\partialΩ)=\partial^{\operatorname{weak}-*}{\bf C}_{2,p'}\,/\,{\bf C}_{2,p'}, \] where ${\bf C}_{2,p'}$ is a certain $L^{p'}$-Carleson space and $p'$ is the Hölder conjugate of $p$. This answers a question considered by Hytönen and Rosén. Inspired by this result and the recently understood characterizations of the $L^p$-solvability of the Dirichlet problem in terms of the Poisson problem by Mourgoglou, Poggi, and Tolsa, we show a novel approximation result: for an arbitrary elliptic operator $L=-\operatorname{div} A\nabla$ with a not necessarily symmetric matrix $A$ of real bounded measurable coefficients, the solution space to the Dirichlet problem with data in $L^p(\partialΩ)$ \[ \left\{\begin{aligned}-\operatorname{div} A\nabla u&=0,\quad&\text{in }&Ω,\\u&=g,\quad&\text{on }&\partialΩ,\end{aligned}\right. \] lies on the weak-$*$ boundary in ${\bf N}_{2,p}$ of the solution space to the Poisson problem \[ \left\{\begin{aligned}-\operatorname{div} A\nabla w&=-\operatorname{div} F,\qquad&\text{in }&Ω,\\ w&=0,\qquad&\text{on }&\partialΩ,\end{aligned}\right. \] with $F\in{\bf C}_{2,p}$, provided that the Dirichlet problem for $L$ with data in $L^p(\partialΩ)$ is solvable in $Ω$. This approximation result is sharp and new even for the Laplacian and on the unit ball.