Unique continuation from the boundary for the spectral fractional Laplacian

Alessandra De Luca (Università di Milano Bicocca)

Abstract: After a brief introduction to the unique continuation principle, I will discuss its validity at the boundary for a class of nonlocal elliptic equations driven by the spectral fractional Laplacian $(-\Delta_D)^s$, defined via fractional powers of the Dirichlet eigenvalues.
More precisely, an extension procedure allows to reduce the problem to a local equation in one higher dimension, posed on a cylinder with a homogeneous Dirichlet condition on the lateral boundary and a non-homogeneous Neumann condition on the base.
For the extended problem, after an odd reflection across the junction between the basis and the lateral surface of the cylinder, enough regularity is available to derive a Pohozaev-type identity and consequently some doubling properties via an Almgren-type monotonicity formula.
Combining this with a blow-up analysis, which gives information on all the admissible vanishing orders of solutions, we get the strong unique continuation property for the nonlocal problem.
Finally, I will also present a new project concerning the Neumann spectral fractional Laplacian $(-\Delta_N)^s$.