A local Hopf lemma and unique continuation for elliptic equations (Q2048622)

This paper is concerned with the unique continuation at the boundary and a local Hopf lemma for solutions of the following elliptic equation \[ \Delta u + \sum_{i=1}^{n} b_i(x) \frac{\partial u}{\partial x_i} + c(x)u = 0 \tag{P} \] where \(b_i(x)\) and \(c(x)\) are real analytic on \(\bar{B}^+_r\) with \(B^+_r = \{x = (x', x_n) \in \mathbb{R}^{n-1}\times \mathbb{R}^1 : |x| < r,x_n > 0\}\). The main result of the paper is as follows: Let \(u\) be a solution of equation (P) in the half ball \(B^+_ r\) and \(C^2\) on \(\bar{B}^+_r\). If \(u\) satisfies that (1) \(u(x', 0) \geq 0\) for \(|x'| \leq r\); (2) the function \(x_n \rightarrow u(0', x_n)\) is flat at \(x_n = 0\); (3) for every positive integer \(N\), the function \(|x'|^{-N} u(x', 0)\) is integrable on \(|x'| \leq r\). Then \(u(x', 0) \equiv 0\) for \(x'\) small.