A Strong Maximum Principle for the fractional Laplace equation with mixed boundary condition

TL;DR

This paper establishes a strong maximum principle for fractional elliptic equations with mixed boundary conditions, extending classical results to the non-local fractional setting and providing comparison results similar to Hopf's Lemma.

Contribution

It introduces a maximum principle for fractional Laplace equations with mixed boundary conditions, extending prior work to non-local operators and establishing comparison results.

Findings

Proves a strong maximum principle for fractional Laplace equations with mixed boundary data.

Provides a comparison result for solutions involving the spectral fractional Laplacian.

Extends classical maximum principles to non-local fractional elliptic problems.

Abstract

In this work we prove a strong maximum principle for fractional elliptic problems with mixed Dirichlet-Neumann boundary data which extends the one proved by J. D\'avila to the fractional setting. In particular, we present a comparison result for two solutions of the fractional Laplace equation involving the spectral fractional Laplacian endowed with homogeneous mixed boundary condition. This result represents a non-local counterpart to a Hopf's Lemma for fractional elliptic problems with mixed boundary data.