The inhomogeneous fractional Dirichlet problem

TL;DR

This work addresses boundary regularity for the inhomogeneous Dirichlet problem for nondegenerate $2s$-stable operators, introducing exterior $C^{1,\text{Dini}}$-type boundary conditions and a generalized Hölder modulus to quantify exterior data regularity. The authors establish that, under these boundary assumptions and with exterior data $g\in C_{\text{ext}}^{\omega}(\Omega^c)$, solutions satisfy a precise modulus of continuity $\sigma(t)= t^s\bigl(1+\int_t^1 \omega(r)/r^{1+s}\,dr\bigr)$ up to the boundary, with a global Hölder-type bound involving the exterior data and the solution tail. They prove sharpness via explicit counterexamples showing that $C^s$ regularity can fail when the Dini condition is violated, and they also discuss one-dimensional cases and sign-changing data that can restore or enhance regularity through cancellation. The results connect with and extend classical boundary-regularity theory (including the fractional Laplacian) and remain robust as $s\to1$, offering a direct analysis of the inhomogeneous problem without subtracting exterior extensions.

Abstract

We study boundary regularity for the inhomogeneous Dirichlet problem for $2s$-stable operators in generalized Hölder spaces. Moreover, we provide explicit counterexamples that showcase the sharpness of our results. Our approach directly addresses the inhomogeneous Dirichlet problem, rather than subtracting an appropriate extension of the exterior data. Even for the fractional Laplacian, our result is new.

Figures (1)

• Figure 1: Illustration of the exterior $C^{1,\text{\normalfont{Dini}}}$-property.

Theorems & Definitions (22)

• theorem 1
• corollary 2
• remark 3
• remark 4
• theorem 6
• remark 7
• lemma 8
• proof
• proposition 9
• proof : Proof of \ref{['prop:regularity_with_modulus']}