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

Abstract

On a bounded domain $\Omega\subset\mathbb R^{n+1}$, $n\geq2$, 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 $L^p(\partial\Omega)$, $p\in(1,\infty)$. We find that \[ ({\bf N}_{2,p})^*={\bf C}_{2,p'}\oplus L^{p'}(\partial\Omega),\qquad\text{and that}\qquad L^{p'}(\partial\Omega)=\partial^{\operatorname{weak}-*}{\bf C}_{2,p'}\,/\,{\bf C}_{2,p'}, \] where ${\bf C}_{2,p'}$ is a certain $L^{p'}$-Carleson space and $p'$ is the H\"older conjugate of $p$. This answers a question considered by Hyt\"onen and Ros\'en. 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 $L^p(\partial\Omega)$ \[ \left\{\begin{aligned}-\operatorname{div} A\nabla u&=0,\quad&\text{in }&\Omega,\\u&=g,\quad&\text{on }&\partial\Omega,\end{aligned}\right. \] lies on the weak-$*$ boundary in ${\bf N}_{2,p}$ of the solution space to the Poisson problem \[ \left\{\begin{aligned}-\operatorname{div} A\nabla w&=-\operatorname{div} F,\qquad&\text{in }&\Omega,\\ w&=0,\qquad&\text{on }&\partial\Omega,\end{aligned}\right. \] with $F\in{\bf C}_{2,p}$, provided that the Dirichlet problem for $L$ with data in $L^p(\partial\Omega)$ is solvable in $\Omega$. This approximation result is sharp and new even for the Laplacian and on the unit ball.