Quantitative uniqueness of solutions to second order elliptic equations with singular potentials in two dimensions

TL;DR

This paper establishes quantitative bounds on the vanishing order and decay rates of solutions to second order elliptic equations with singular potentials in two dimensions, using new Carleman estimates.

Contribution

It introduces a novel $L^p - L^q$ Carleman estimate for the Laplacian in $ r^2$, enabling precise vanishing order and decay rate bounds.

Findings

Derived lower bounds for solutions on small balls based on Lebesgue norms.

Established decay rate estimates at infinity for solutions.

Introduced a new Carleman estimate for the Laplacian in two dimensions.

Abstract

In this article, we study the vanishing order of solutions to second order elliptic equations with singular lower order terms in the plane. In particular, we derive lower bounds for solutions on arbitrarily small balls in terms of the Lebesgue norms of the lower order terms for all admissible exponents. Then we show that a scaling argument allows us to pass from these vanishing order estimates to estimates for the rate of decay of solutions at infinity. Our proofs rely on a new $L^p - L^q$ Carleman estimate for the Laplacian in $\mathbb{R}^2$.