The Dirichlet problem as the boundary of the Poisson problem: A sharp approximation result

Abstract

On a bounded domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^{n+1}$, $n\ge 2$, satisfying the corkscrew condition and with Ahlfors regular boundary, we characterize the dual space to the space ${\mathbf{N}}_{2,p}$ of functions $u$ whose Kenig-Pipher modified non-tangential maximal operator ${\mathcal{N}}_2(u)$ lies in $L^p(\partial \hat{I}\text{c◯})$, $p\in (1,\infty )$. 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 ${\mathbf{C}}_{2,p^{\prime }}$ is a certain $L^{p^{\prime }}$-Carleson space and $p^{\prime }$ 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 $L=-\mathrm{div}A\nabla$ with a not necessarily symmetric matrix $A$ of real bounded measurable coefficients, the solution space to the Dirichlet problem with data in $L^p(\partial \hat{I}\text{c◯})$ \[ \left\{\begin{aligned}-\operatorname{div} A\nabla u&=0,\quad&\text{in }&Ω,\\u&=g,\quad&\text{on }&\partialΩ,\end{aligned}\right. \] lies on the weak-$\ast$ boundary in ${\mathbf{N}}_{2,p}$ 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 $F\in {\mathbf{C}}_{2,p}$, provided that the Dirichlet problem for $L$ with data in $L^p(\partial \hat{I}\text{c◯})$ is solvable in $\hat{I}\text{c◯}$. This approximation result is sharp and new even for the Laplacian and on the unit ball.

25 pages