Unique continuation principles for a higher order fractional Laplace equationThe authors are partially supported by the INDAM-GNAMPA 2018 grant ‘Formula di monotonia e applicazioni: problemi frazionari e stabilità spettrale rispetto a perturbazioni del dominio’. V Felli is partially supported by the PRIN 2015 grant ‘Variational methods, with applications to problems in mathematical physics and geometry’.

Abstract

In this paper we prove the strong unique continuation principle and the unique continuation from sets of positive measure for solutions of a higher order fractional Laplace equation in an open domain. Our proofs are based on the Caffarelli–Silvestre (2007 Commun. PDE 32 1245–60) extension method combined with an Almgren type monotonicity formula. The corresponding extended problem is formulated as a system of two second order equations with singular or degenerate weights in a half-space, for which asymptotic estimates are derived by a blow-up analysis.