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.

De Luca, A., Felli, V., Siclari, G. (2023). Strong Unique Continuation from the Boundary for the Spectral Fractional Laplacian. ESAIM. COCV, 29 [10.1051/cocv/2023045].

Strong Unique Continuation from the Boundary for the Spectral Fractional Laplacian

De Luca, Alessandra;Felli, Veronica
;
Siclari, Giovanni
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.
Articolo in rivista - Articolo scientifico
Boundary behaviour of solutions; Monotonicity formula; Spectral fractional Laplacian; Unique continuation;
English
28-giu-2023
2023
29
50
open
De Luca, A., Felli, V., Siclari, G. (2023). Strong Unique Continuation from the Boundary for the Spectral Fractional Laplacian. ESAIM. COCV, 29 [10.1051/cocv/2023045].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/427358
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