Extension problem for the fractional parabolic Lamé operator and unique continuation

Abstract

In this paper, we introduce and analyse an explicit formulation of fractional powers of the parabolic Lamé 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 Lamé operator to another system that resembles the extension problem of the fractional heat operator. Finally in the case when $s\ge 1/2$, 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 ${\mathbb{H}}^s\text{u}=V\text{u}$.

a few bibliographical omissions in the previous version corrected