A local Hopf lemma and unique continuation for elliptic equations
type: Item
instance of: scholarly article
title: A local Hopf lemma and unique continuation for elliptic equations
zbMATH Open document ID: 1479.35249
published in: Advances in Mathematics
publication date: 2021-08-23T00:00:00Z
review text: 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.
reviewed by: Huansong Zhou
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35B60
Mathematics Subject Classification ID: 31B25
Mathematics Subject Classification ID: 30B40
zbMATH DE Number: 7384222
zbMATH Keywords: unique continuation
zbMATH Keywords: Hopf lemma
zbMATH Keywords: Poisson kernel
author: S. Berhanu
MaRDI profile type: Publication
Wikidata QID: Q124872374
full work available at URL: j.aim.2021.107912
OpenAlex ID: W3189599306
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DOI: 10.1016/J.AIM.2021.107912
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description: scientific article; zbMATH DE number 7384222