Unique continuation from conical boundary points for fractional equations

TL;DR

This paper investigates the behavior of solutions to fractional elliptic equations near conical boundary points, establishing asymptotics, a unique continuation principle, and new analytical tools for corner singularities.

Contribution

It introduces a novel method combining smoothing, approximation, and spectral analysis to derive asymptotics and unique continuation for fractional equations at conical boundaries.

Findings

Asymptotic behavior characterized at conical boundary points

Established a strong unique continuation principle

Developed a Pohozaev type inequality for fractional equations

Abstract

We provide fine asymptotics of solutions of fractional elliptic equations at boundary points where the domain is locally conical; that is, corner type singularities appear. Our method relies on a suitable smoothing of the corner singularity and an approximation scheme, which allow us to provide a Pohozaev type inequality. Then, the asymptotics of solutions at the conical point follow by an Almgren type monotonicity formula, blow-up analysis and Fourier decomposition on eigenspaces of a spherical eigenvalue problem. A strong unique continuation principle follows as a corollary.