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

Abstract: On a bounded domain $Ω\subset\mathbb R<sup>{n+1}$, $n\geq2$, satisfying the corkscrew condition and with Ahlfors regular boundary, we characterize the dual space to the space ${\bf N}<em>{2,p}$ of functions $u$ whose Kenig-Pipher modified non-tangential maximal operator $\mathcal N_2(u)$ lies in $L<sup>p(\partialΩ)$, $p\in(1,\infty)$. 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 ${\bf C}</em>{2,p&#39;}$ is a certain $L<sup>{p&#39;}$-Carleson space and $p&#39;$ 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 $L<sup>p$-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 $L<sup>p(\partialΩ)$ [ \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 ${\bf N}<em>{2,p}$ 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 $F\in{\bf C}</em>{2,p}$, provided that the Dirichlet problem for $L$ with data in $L<sup>p(\partialΩ)$ is solvable in $Ω$. This approximation result is sharp and new even for the Laplacian and on the unit ball.

• The paper proves the sharp duality formula $({\bf N}_{r,p})^*={\bf C}_{r',p'}\oplus L^{p'}(\partial\Omega)$ for $r<\infty$, identifying boundary functionals with $L^{p'}$ data.
• The paper constructs compactly supported Lipschitz forcing fields with uniformly controlled Carleson norm whose zero-boundary Poisson solutions converge strongly locally and weak-* in ${\bf N}_{2,p}$ to every solvable Dirichlet solution.
• The paper shows that weak-* convergence is optimal for nonzero boundary data and interprets Dirichlet solutions as lying on the weak-* boundary of the Poisson-Dirichlet solution space, even for nonsymmetric real elliptic operators.

Overview and motivation

This paper by Mourgoglou and Poggi establishes two related results concerning the relationship between the $L^p$-Dirichlet problem $(\operatorname{D}_p^L)$ for a real elliptic operator $L=-\operatorname{div}A\nabla$ and the associated inhomogeneous Poisson-Dirichlet problem $(\operatorname{PD}_p^L)$ with data in the Carleson space ${\bf C}_{2,p}$. The first result characterizes the dual space of ${\bf N}_{r,p}$, answering a question posed by Hytönen and Rosén; the second is an approximation theorem showing that solutions to the homogeneous Dirichlet problem lie on the weak-$*$ boundary of the solution space of the Poisson-Dirichlet problem. The setting is a bounded domain $\Omega\subset\mathbb{R}^{n+1}$, $n\geq 2$, satisfying the corkscrew condition with $n$-Ahlfors regular boundary, and $(\operatorname{D}_p^L)$0 a real, not necessarily symmetric, bounded measurable strongly elliptic matrix.

The motivation comes from the recent equivalence proved in [(2602.07560)'s cited work mpt25]: solvability of $(\operatorname{D}_p^L)$1 is quantitatively equivalent to solvability of $(\operatorname{D}_p^L)$2, where the latter asks for

$(\operatorname{D}_p^L)$3

for solutions of $(\operatorname{D}_p^L)$4 with zero boundary data. That equivalence, however, was established at the level of solution operators via harmonic measure and Green's function techniques, leaving open how individual solutions relate. The present paper addresses precisely this granular question.

The dual characterization of $(\operatorname{D}_p^L)$5

The first main theorem identifies the dual of $(\operatorname{D}_p^L)$6 — defined here as the completion of $(\operatorname{D}_p^L)$7 under the norm $(\operatorname{D}_p^L)$8, a stricter definition than that of Hytönen–Rosén and Mourgoglou–Poggi–Tolsa, chosen so that every element has a well-defined trace on $(\operatorname{D}_p^L)$9. For corkscrew domains with Ahlfors regular boundary (bounded or with unbounded boundary), $L=-\operatorname{div}A\nabla$0, $L=-\operatorname{div}A\nabla$1:

$L=-\operatorname{div}A\nabla$2

meaning every functional $L=-\operatorname{div}A\nabla$3 admits a unique representation $L=-\operatorname{div}A\nabla$4 with $L=-\operatorname{div}A\nabla$5 and $L=-\operatorname{div}A\nabla$6. Combined with the earlier quantitative quasi-duality (which showed $L=-\operatorname{div}A\nabla$7 and forced the quotient $L=-\operatorname{div}A\nabla$8 into the weak-$L=-\operatorname{div}A\nabla$9 boundary of $(\operatorname{PD}_p^L)$0), this yields the sharp identification

$(\operatorname{PD}_p^L)$1

The proof proceeds by Riesz representation: a functional on $(\operatorname{PD}_p^L)$2 restricts to a measure $(\operatorname{PD}_p^L)$3 on $(\operatorname{PD}_p^L)$4, which must be shown to decompose as $(\operatorname{PD}_p^L)$5. Absolute continuity of $(\operatorname{PD}_p^L)$6 against Lebesgue measure on compact subsets of $(\operatorname{PD}_p^L)$7 follows from a covering argument bounding $(\operatorname{PD}_p^L)$8; the Carleson membership of $(\operatorname{PD}_p^L)$9 follows from truncation, Fatou's lemma, and the duality estimates. The boundary part is subtler: absolute continuity of ${\bf C}_{2,p}$0 with respect to ${\bf C}_{2,p}$1 is obtained by testing against approximations ${\bf C}_{2,p}$2 of indicators of relatively open sets, extending them to ${\bf C}_{2,p}$3 via the Varopoulos extension, and using the quantitative extension bounds. A notable point is that pointwise (not merely ${\bf C}_{2,p}$4-a.e.) convergence of the approximants is required, since no relation between ${\bf C}_{2,p}$5 and ${\bf C}_{2,p}$6 is assumed a priori.

Two caveats are stated explicitly. First, the identity fails for ${\bf C}_{2,p}$7: for the half-space, ${\bf C}_{2,p}$8 is the space of Carleson measures, and e.g. a Dirac mass at an interior point defines a bounded functional not admitting the decomposition above. Second, for the larger space ${\bf C}_{2,p}$9 used in prior work (whose elements need not have boundary traces), the decomposition still holds on ${\bf N}_{r,p}$0 but cannot be extended uniquely to all of ${\bf N}_{r,p}$1; Hahn-Banach extensions of boundary functionals exist but lose uniqueness.

The approximation theorem

The second main result states that if ${\bf N}_{r,p}$2 is solvable and ${\bf N}_{r,p}$3 solves the Dirichlet problem with data ${\bf N}_{r,p}$4, then there exist ${\bf N}_{r,p}$5 such that:

• Uniform Carleson control: ${\bf N}_{r,p}$6, with ${\bf N}_{r,p}$7 depending only on dimension, ellipticity, and geometric constants.
• Convergence of Poisson solutions: writing ${\bf N}_{r,p}$8 for the solution of ${\bf N}_{r,p}$9, $*$0 on $*$1, one has $*$2 strongly in $*$3, strongly in $*$4, and weak-$*$5 in $*$6 — and this last convergence cannot be improved to weak convergence unless $*$7.

This result is new even for the Laplacian on the unit ball. It may appear counterintuitive since Dirichlet solutions have nontrivial boundary traces while all $*$8 vanish identically on $*$9; this obstruction is exactly what precludes upgrading weak-$\Omega\subset\mathbb{R}^{n+1}$0 to weak convergence, because $\Omega\subset\mathbb{R}^{n+1}$1 by the dual characterization, and $\Omega\subset\mathbb{R}^{n+1}$2 for every such $\Omega\subset\mathbb{R}^{n+1}$3.

The construction is explicit modulo regularization: for Lipschitz data $\Omega\subset\mathbb{R}^{n+1}$4 with Varopoulos extension $\Omega\subset\mathbb{R}^{n+1}$5, one sets $\Omega\subset\mathbb{R}^{n+1}$6 where $\Omega\subset\mathbb{R}^{n+1}$7 is a cutoff supported in a $\Omega\subset\mathbb{R}^{n+1}$8-neighborhood of the boundary. The key estimate is the pointwise Carleson bound

$\Omega\subset\mathbb{R}^{n+1}$9

proved via a Besicovitch-covering annular decomposition of the strip supporting $n\geq 2$0, reducing to the Hardy–Littlewood maximal function of $n\geq 2$1, which the Varopoulos extension controls by $n\geq 2$2. The $n\geq 2$3 are then obtained by density of $n\geq 2$4 in $n\geq 2$5 with error $n\geq 2$6.

For general $n\geq 2$7, one approximates by Lipschitz data $n\geq 2$8, applies the Lipschitz-case construction, and diagonalizes along indices $n\geq 2$9. The weak-$n$0 convergence argument tests against arbitrary $n$1: for smooth compactly supported $n$2, integration by parts against the adjoint solution $n$3 of $n$4 reduces matters to the conormal derivative pairing $n$5 — which vanishes identically since $n$6 — plus terms controlled by the solvability of the Poisson-Regularity problem $n$7, itself a consequence of $n$8 via [mpt25]. Density of $n$9 in $(\operatorname{D}_p^L)$00 then handles general test functions. Notably, the proof avoids harmonic measure and Green's function arguments entirely; its main analytic input beyond [mpt25] is the state-of-the-art Varopoulos extension from [mz25], which is what permits the high geometric generality (the extension being nonlinear, density arguments require care).

Structural interpretation

Combining the two results yields the paper's conceptual punchline, formalized as a corollary: letting $(\operatorname{D}_p^L)$01 and $(\operatorname{D}_p^L)$02 denote the solution spaces of $(\operatorname{D}_p^L)$03 and $(\operatorname{D}_p^L)$04 respectively,

$(\operatorname{D}_p^L)$05

mirroring the identity $(\operatorname{D}_p^L)$06 at the data level. In this precise sense, the Dirichlet problem "lives on the boundary" of the Poisson-Dirichlet problem: the data spaces $(\operatorname{D}_p^L)$07 and $(\operatorname{D}_p^L)$08 appearing in the dual decomposition correspond exactly to the data classes of the two boundary value problems, and this structure lifts to the level of individual solutions.

Limitations and open questions

Several restrictions are acknowledged. The dual characterization requires $(\operatorname{D}_p^L)$09 and fails at $(\operatorname{D}_p^L)$10 for structural reasons (Carleson measures). The approximation theorem is stated for bounded domains and the exponent $(\operatorname{D}_p^L)$11 in $(\operatorname{D}_p^L)$12 and $(\operatorname{D}_p^L)$13; the authors note these can be relaxed ($(\operatorname{D}_p^L)$14 in general, $(\operatorname{D}_p^L)$15 if $(\operatorname{D}_p^L)$16 is locally Lipschitz) with suitable modifications, and boundedness is a convenience rather than essential. No analogue holds for continuous boundary data: $(\operatorname{D}_p^L)$17 admits no predual (hence no weak-$(\operatorname{D}_p^L)$18 topology), and the uniform norm is too strong to control natural candidates for the approximating data, so even the analogue of the uniform Carleson estimate is unclear. Whether the inclusion $(\operatorname{D}_p^L)$19 can be sharpened to equality, or extended to other boundary value problems (Regularity, Neumann) and rougher or unbounded settings, remains open.

Conclusion

The paper resolves the Hytönen–Rosén duality question by proving $(\operatorname{D}_p^L)$20 on corkscrew domains with Ahlfors regular boundary, and leverages this structure to prove a sharp, globally convergent approximation of Dirichlet solutions by zero-boundary-data Poisson solutions with uniformly controlled Carleson data — valid for arbitrary real elliptic operators without symmetry assumptions and new already for the Laplacian on the ball. Together these results give a mechanism-free, kernel-free explanation of the Dirichlet/Poisson equivalence, identifying the former as the weak-$(\operatorname{D}_p^L)$21 boundary of the latter at the level of both data spaces and individual solutions.