Strong unique continuation from the boundary for the spectral fractional Laplacian

TL;DR

This paper establishes strong unique continuation properties for solutions to spectral fractional Laplacian equations by analyzing boundary behavior through an extension problem and monotonicity formulas.

Contribution

It introduces a novel approach using an extension procedure and Almgren-type monotonicity to classify boundary asymptotics and prove unique continuation for spectral fractional Laplacian.

Findings

Classification of local asymptotic profiles at boundary points

Proof of strong unique continuation property for fractional Laplacian solutions

Development of an extension method linking boundary behavior to interior properties

Abstract

We investigate unique continuation properties and asymptotic behaviour at boundary points for solutions to a class of elliptic equations involving the spectral fractional Laplacian. An extension procedure leads us to study a degenerate or singular equation on a cylinder, with a homogeneous Dirichlet boundary condition on the lateral surface and a non homogeneous Neumann condition on the basis. For the extended problem, by an Almgren-type monotonicity formula and a blow-up analysis, we classify the local asymptotic profiles at the edge where the transition between boundary conditions occurs. Passing to traces, an analogous blow-up result and its consequent strong unique continuation property is deduced for the nonlocal fractional equation.