Mathematical Research Letters

Volume 22 (2015)

Number 4

Quantitative uniqueness estimates for second order elliptic equations with unbounded drift

Pages: 1159 – 1175



Carlos Kenig (Department of Mathematics, University of Chicago, Chicago, Illinois, U.S.A.)

Jenn-Nan Wang (Institute of Applied Mathematical Sciences, NCTS, National Taiwan University, Taipei, Taiwan)


In this paper we derive quantitative uniqueness estimates at infinity for solutions to an elliptic equation with unbounded drift in the plane. More precisely, let $u$ be a real solution to $\Delta u+W\cdot\nabla u=0$ in $\mathbf{R}^2$, where $W$ is real vector and $\|W\|_{L^p(\mathbf{R}^2)}\le K$ for $2\le p<\infty$. Assume that $u$ satisfies certain a priori assumption at $0$. For $2<p<\infty$, if $\|u\|_{L^{\infty}(\mathbf{R}^2)}\le C_0$, then $u$ satisfies the following asymptotic estimates at $R\gg 1$\[\inf_{|z_0|=R}\sup_{|z-z_0|<1}|u(z)|\ge \exp(-C_1R^{1-2/p}\log R),\]where $C_1>0$ depends on $p, K, C_0$. For $p=2$, if $|u(z)|\le |z|^m$ for $|z|>1$ with some $m>0$, then\[\inf_{|z_0|=R}\sup_{|z-z_0|<1}|u(z)|\ge C_2\exp(-C_3(\log R)^2),\]where $C_2>0$ depends on $m$ and $C_3$ depends on $m, K$. Using the scaling argument in [BK05], these quantitative estimates are easy consequences of estimates of the maximal vanishing order for solutions of the local problem. The estimate of the maximal vanishing order is a quantitative form of the strong unique continuation property.

Accepted 22 February 2015

Published 24 July 2015