Strong unique continuation and local asymptotics at the boundary for fractional elliptic equations

TL;DR

This paper investigates the boundary behavior of solutions to fractional elliptic equations, establishing strong unique continuation principles and classifying asymptotic profiles using blow-up analysis and monotonicity formulas.

Contribution

It introduces a novel approach combining blow-up techniques and Almgren type monotonicity formulas to analyze boundary asymptotics and unique continuation for fractional elliptic equations.

Findings

Classification of boundary asymptotics for solutions

Proof of strong unique continuation from boundary points

Development of new monotonicity formulas for fractional equations

Abstract

We study local asymptotics of solutions to fractional elliptic equations at boundary points, under some outer homogeneous Dirichlet boundary condition. Our analysis is based on a blow-up procedure which involves some Almgren type monotonicity formulae and provides a classification of all possible homogeneity degrees of limiting entire profiles. As a consequence, we establish a strong unique continuation principle from boundary points.