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# Mathematical Research Letters

## Volume 22 (2015)

### Number 4

### Quantitative uniqueness estimates for second order elliptic equations with unbounded drift

Pages: 1159 – 1175

DOI: http://dx.doi.org/10.4310/MRL.2015.v22.n4.a10

#### Authors

#### Abstract

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.