Unique continuation principles for a higher order fractional Laplace equation

TL;DR

This paper establishes strong and measure-based unique continuation principles for higher order fractional Laplace equations using extension methods and monotonicity formulas, advancing understanding of solution behavior.

Contribution

It introduces novel techniques combining extension methods and monotonicity formulas to prove unique continuation for higher order fractional Laplace equations.

Findings

Proves strong unique continuation principle.

Establishes unique continuation from sets of positive measure.

Develops asymptotic estimates for extended systems.

Abstract

In this paper we prove strong unique continuation principle and 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 extension method combined with an Almgren type monotonicity formula. The corresponding extended problem is formulated as a systems of two second order equations with singular or degenerate weights in a half-space, for which asymptotics estimates are derived by a blow-up analysis.