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

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 $L^{\mathrm{\infty }}$ 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 $L^{\mathrm{\infty }}$ norm by only a local $L^{\mathrm{\infty }}$ 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.