Quantitative boundary Hölder estimates for the inhomogeneous Poisson problem through a probabilistic approach
Iulian Cîmpean, Ionel Popescu, Arghir Zarnescu
TL;DR
The paper develops quantitative boundary Hölder estimates for the inhomogeneous Poisson problem in bounded domains $D\subset \mathbb{R}^d$ by a probabilistic representation tied to Brownian exit times. It first shows that boundary Hölder regularity of the solution $u$ is controlled by the Hölder seminorm of the boundary data $g$, the $L^\gamma(D)$ norm of the source $f$ with $\gamma>d/2$, and the exit-time moment $v_{D,\alpha/2}$, then derives explicit bounds for $v_{D,\alpha/2}$ in domains with exterior ball, cone, or wedge geometries, yielding concrete, explicit constants for the behavior of $u$, its gradient, and the Green function near the boundary. The authors present two analytic routes—a reverse-doubling oscillation method and an Itô/barrier approach—to obtain sharp exit-time estimates, and apply them across geometric classes to produce dimension-explicit Hölder exponents and constants. Consequences include explicit boundary gradient bounds and Green-function estimates, with potential impact on high-dimensional PDE approximations and neural-network-based solvers through explicit regularity information. The framework is posed to extend to more general elliptic operators via a martingale-problem perspective, offering a probabilistic pathway to quantitative boundary regularity in complex domains.
Abstract
In this paper we derive quantitative boundary Hölder estimates, with explicit constants, for the inhomogeneous Poisson problem in a bounded open set $D\subset \mathbb{R}^d$. Our approach has two main steps: firstly, we consider an arbitrary $D$ as above and prove that the boundary $α$-Hölder regularity of the solution the Poisson equation is controlled, with explicit constants, by the Hölder seminorm of the boundary data, the $L^ γ$-norm of the forcing term with $γ>d/2$, and the $α/2$-moment of the exit time from $D$ of the Brownian motion. Secondly, we derive explicit estimates for the $α/2$-moment of the exit time in terms of the distance to the boundary, the regularity of the domain $D$, and $α$. Using this approach, we derive explicit estimates for the same problem in domains satisfying exterior ball conditions, respectively exterior cone/wedge conditions, in terms of simple geometric features. As a consequence we also obtain explicit constants for pointwise estimates for the Green function and for the gradient of the solution. The obtained estimates can be employed to bypass the curse of high dimensions when aiming to approximate the solution of the Poisson problem using neural networks, obtaining polynomial scaling with dimension, which in some cases can be shown to be optimal.