Large harmonic functions for fully nonlinear fractional operators

TL;DR

This paper investigates the existence, uniqueness, and boundary behavior of harmonic functions related to fully nonlinear fractional operators, extending classical results to nonlocal, nonlinear contexts with new boundary blow-up profiles.

Contribution

It introduces a method using viscosity solutions and Perron's method to construct boundary blow-up solutions for fully nonlinear fractional operators, including the linear fractional Laplacian.

Findings

Constructed harmonic functions with prescribed boundary blow-up profiles.

Provided boundary blow-up rates and gradient estimates.

Extended results to nonlinear fractional operators, including the linear case.

Abstract

We study existence, uniqueness and boundary blow-up profile for fractional harmonic functions on a bounded smooth domain $\Omega \subset \mathbb R^N$. We deal with harmonic functions associated to uniformly elliptic, fully nonlinear nonlocal operators, including the linear case $$ (-\Delta)^s u = 0 \quad \mbox{in} \ \Omega, $$ where $(-\Delta)^s$ denotes the fractional Laplacian of order $2s \in (0,2)$. We use the viscosity solution's theory and Perron's method to construct harmonic functions with zero exterior condition in $\bar \Omega^c$, and boundary blow-up profile $$ \lim_{x\to x_0, x \in \Omega}\mathrm{dist}(x, \partial \Omega)^{1-s}u(x)=h(x_0), \quad \mbox{for all} \quad x_0\in \partial \Omega, $$ for any given boundary data $h \in C(\partial \Omega)$. Our method allows us to provide blow-up rate for the solution and its gradient estimates. Results are new even in the linear case.