Uniqueness of solutions to an elliptic inequality with rapid decay at infinity

TL;DR

This paper proves the uniqueness of solutions to a specific elliptic inequality in an exterior domain, showing that solutions with certain rapid decay at infinity must be identically zero, using Carleman estimates.

Contribution

It establishes new uniqueness results for elliptic inequalities with rapid decay, employing Carleman estimates tailored to the decay rates in different coordinate directions.

Findings

Solutions with prescribed decay rates are identically zero.

Decay conditions depend on constants in the inequality.

Carleman estimates are effective for proving uniqueness.

Abstract

We consider an elliptic differential inequality: $\vert \Delta u(x) \vert \le C_0(\YYYY^{-\gamma}\vert u(x)\vert + \YYYY^{-\theta}\vert \nabla u(x)\vert)$ in an exterior domain $\R^n \setminus \ooo{U}$, where $U$ is a simply connected bounded domain $U$, $x := (y,z) \in \R^n$ with $y \in \R^m$ and $z\in \R^{n-m}$ for given $m\in \{ 1, ..., n\}$, and $\gamma, \theta \in \R$ are constants. We assume that $u(x)$ decays with exponential rate in the $y$-coordinates and polynomial rate in the $z$-coordinates as $\vert x\vert \to \infty$. We prove that if decay rates of $u$ satisfy certain conditions related to the constants $\gamma, \theta \in \R$, then $u\equiv 0$ in $\UUUUU$. The key is a Carleman estimate with typical cut-off arguments.