Extension problem for the fractional parabolic Lam\'e operator and unique continuation

TL;DR

This paper introduces a fractional parabolic Lamé operator, studies its extension problem, and establishes a unique continuation property for solutions when the fractional order is at least 1/2.

Contribution

It provides an explicit formulation of fractional powers of the parabolic Lamé operator and analyzes the associated extension problem, including regularity and unique continuation results.

Findings

Explicit formulation of fractional parabolic Lamé operator

Regularity properties of solutions to the extension problem

Unique continuation result for solutions with fractional order ≥ 1/2

Abstract

In this paper, we introduce and analyse an explicit formulation of fractional powers of the parabolic Lam\'e operator $\mathbb{H}$ and we then study the extension problem associated to such non-local operators. We also study the various regularity properties of solutions to such an extension problem via a transformation which reduces the extension problem for the parabolic Lam\'e operator to another system that resembles the extension problem of the fractional heat operator. Finally in the case when $s \geq 1/2$, by proving a conditional doubling property for solutions to the corresponding reduced system followed by a blowup argument, we establish a space-like strong unique continuation result for $\mathbb{H}^s \textbf{u}=V\textbf{u}$.