Strong unique continuation from the boundary for the spectral fractional laplacian

¹ Dipartimento di Scienze Molecolari e Nanosistemi, Università Ca’ Foscari Venezia, Via Torino 155, 30172 Venezia Mestre, Italy
² Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, Via Cozzi 55, 20125 Milano, Italy

^* Corresponding author: veronica.felli@unimib.it

Received: 23 February 2023
Accepted: 29 May 2023

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.