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

TL;DR

This paper investigates boundary regularity and a priori estimates for solutions to fractional equations on unbounded domains, introducing local regularity results to address limitations of global norm reliance.

Contribution

It develops local boundary H"older regularity results for fractional equations, enabling new a priori estimates in unbounded domains.

Findings

Established local boundary H"older regularity for solutions.

Derived a priori estimates for nonlinear fractional equations.

Extended regularity results to unbounded domains.

Abstract

In this paper, we study the boundary H\"older regularity for solutions to the fractional Dirichlet problem in unbounded domains with boundary \begin{equation*} \begin{cases} (-\Delta)^s u(x) = g(x),&\text{in } \Omega, u(x)=0, &\text{in } \Omega^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 H\"older 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.