Boundary regularity and a priori estimates for fractional equations on unbounded domains

TL;DR

The paper develops local boundary Hölder regularity results for the fractional Laplacian on unbounded domains by replacing global $L^{\infty}$ data with local bounds, enabling rigorous blow-up/rescaling analysis. It proves that nonnegative solutions to $(-\Delta)^s u=g$ in a locally $C^{1,1}$ domain satisfy $u\in C^s$ up to the boundary with quantitative bounds expressed in local data, via a decomposition into a potential and a harmonic part and a precise decay near the boundary. Building on these regularity results, it derives a priori bounds for nonnegative solutions to nonlinear fractional equations $(-\Delta)^s u=f(x,u)$ on unbounded domains, under growth and asymptotic conditions on $f$, by employing a blow-up argument and Liouville-type results. An appendix extends convergence theories for nonlocal equations to unbounded domains with boundary, establishing a boundary-adjusted limit for sequences of fractional Laplacians. Overall, the work advances boundary regularity theory for nonlocal problems on unbounded domains and provides a robust framework for a priori estimates in nonlinear settings.

Abstract

In this paper, we study the boundary Hölder regularity for solutions to the fractional Dirichlet problem in unbounded domains with boundary \begin{equation*} \begin{cases} (-Δ)^s u(x) = g(x),&\text{in } Ω, u(x)=0, &\text{in } Ω^c. \end{cases} \end{equation*} Existing results rely on the global $L^{\infty}$ norm of solutions to control their boundary $C^s$ norm, which is insufficient for blow-up and rescaling analysis to obtain a priori estimates in unbounded domains. To overcome this limitation, we first derive a local version of boundary Hölder regularity for nonnegative solutions in which we replace the global $L^{\infty}$ norm by only a local $L^{\infty}$ norm. Then as an important application, we establish a priori estimates for nonnegative solutions to a family of nonlinear equations on unbounded domains with boundaries.

Figures (1)

• Figure 1: The geometry near a boundary point $x_0$.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Remark 1.1
• Theorem 1.3
• Proposition 2.1: Rosoton2014Dirichlet
• Theorem 2.1
• proof
• Proposition 2.2: Interior Hölder regularity
• Lemma 2.3
• proof : Proof of Lemma \ref{['global holder lemma']}