Uniqueness of solutions to an elliptic inequality with rapid decay at infinity
F. Golgeleyen, O. Y. Imanuvilov, M. Yamamoto
TL;DR
This work establishes that solutions to an elliptic inequality with anisotropic decay in an exterior domain must vanish under appropriately strong decay conditions. The authors develop a Carleman estimate with a weight depending only on the coordinates in which the solution decays exponentially, identify a critical decay rate through the parameter $\alpha$, and show that if $|u(x)| \le C g(|z|) e^{-C_1|y|^{2\beta}}$ with $\beta>\alpha$, then $u \equiv 0$. Two main results are proved: Theorem 1 handles fast exponential decay in $m$ components and polynomial decay in the rest, while Theorem 2 specializes to the case $m=1$; both rely on a detailed Carleman framework and cut-off arguments. The paper also provides supplementary uniqueness results for more general elliptic equations and proves the key Carleman lemma (Lemma P1) that drives the analysis. Overall, the results contribute new coordinate-wise decay criteria for uniqueness, enriching the landscape of Landis–Meshkov-type questions and their applications to inverse problems.
Abstract
We consider an elliptic differential inequality: $\vert Δu(x) \vert \le C_0(\YYYY^{-γ}\vert u(x)\vert + \YYYY^{-θ}\vert \nabla u(x)\vert)$ 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 $γ, θ\in \R$ 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 $γ, θ\in \R$, then $u\equiv 0$ in $\UUUUU$. The key is a Carleman estimate with typical cut-off arguments.