On the strong unique continuation property for the Dirac operator

TL;DR

This paper determines the optimal constant for the strong unique continuation property of the Dirac operator near the origin, extends results to different decay rates, and proves continuation from infinity.

Contribution

It establishes the sharp constant for the Dirac inequality, explores unique continuation for various decay rates, and proves continuation from infinity.

Findings

The constant 1/2 is optimal in the Dirac inequality.

Unique continuation holds for decay rates with gamma > 1.

Strong unique continuation is valid from infinity.

Abstract

In [DO99,KY99], the strong unique continuation property from the origin is established for $H_{loc}^1$-solutions to the massless Dirac differential inequality $|{D}_n u | \leq \frac{C}{|x|}|u|$, in dimension $n\geq 2$ and with $C<\frac12$. We show that $\frac12$ is the largest possibile constant in this result, providing an example in $\mathbb{R}^2$ of a (non-trivial) solution of the inequality. Also, we show properties of unique continuation from the origin for solutions to the inequality $|D_n u | \leq \frac{C}{ |x|^{\gamma}}|u|$, for $\gamma>1$, $C>0$. Finally, we establish the strong unique continuation property for the Dirac operator from the point at infinity.