The Dirichlet problem as the boundary of the Poisson problem: A sharp approximation result
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908819364249600 |
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| author | Mourgoglou, Mihalis Poggi, Bruno |
| author_facet | Mourgoglou, Mihalis Poggi, Bruno |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_07560 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Dirichlet problem as the boundary of the Poisson problem: A sharp approximation result Mourgoglou, Mihalis Poggi, Bruno Analysis of PDEs 35J25, 35J05 (Primary) 35J15, 35J67, 35B30, 42B37 (Secondary) 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. |
| title | The Dirichlet problem as the boundary of the Poisson problem: A sharp approximation result |
| topic | Analysis of PDEs 35J25, 35J05 (Primary) 35J15, 35J67, 35B30, 42B37 (Secondary) |
| url | https://arxiv.org/abs/2602.07560 |