On the Harnack inequality for antisymmetric $s$-harmonic functions

TL;DR

This paper establishes the Harnack inequality for antisymmetric fractional harmonic functions, enabling analysis of symmetry and stability in solutions to fractional PDEs with zero-th order terms.

Contribution

It introduces the first Harnack inequality for antisymmetric $s$-harmonic functions and develops related boundary and interior estimates.

Findings

Proves Harnack inequality for antisymmetric $s$-harmonic functions

Derives a new mean value formula for these functions

Provides tools for symmetry and stability analysis in fractional PDEs

Abstract

We prove the Harnack inequality for antisymmetric $s$-harmonic functions, and more generally for solutions of fractional equations with zero-th order terms, in a general domain. This may be used in conjunction with the method of moving planes to obtain quantitative stability results for symmetry and overdetermined problems for semilinear equations driven by the fractional Laplacian. The proof is split into two parts: an interior Harnack inequality away from the plane of symmetry, and a boundary Harnack inequality close to the plane of symmetry. We prove these results by first establishing the weak Harnack inequality for super-solutions and local boundedness for sub-solutions in both the interior and boundary case. En passant, we also obtain a new mean value formula for antisymmetric $s$-harmonic functions.