Two theorems on removable singularities
type: Item
instance of: scholarly article
title: Two theorems on removable singularities
zbMATH Open document ID: 0780.35014
author: I. Yu. Chesnokov
published in: Moscow University Mathematics Bulletin
publication date: 1993-03-11T00:00:00Z
review text: The author proves two theorems on removable singularities for linear partial differential equations having a special structure with respect to groups of variables. Let \(\Omega\) be a domain in the space of variables \((x,y)\in\mathbb{R}^ m \oplus \mathbb{R}^ n\). Theorem 1: Let \(P=\Delta_ x^ k+ P_ 1(x,y, \partial_ x, \partial_ y)\), where \(\Delta_ x\) is the Laplacian with respect to the \(x\) variables, \(k\) is a positive integer, \(P_ 1\) has smooth coefficients, and \(\deg_ x P_ 1<2k\). Suppose \(Pu=0\) in \(\Omega\setminus A\), where \(A\) is a closed subset of \(\Omega\), and suppose \(u\in W^{2\ell}_{\infty,\text{loc}} (\Omega)\), where \(\ell\) is a positive integer, and \(\ell<k\). Let \(\gamma=m-2 (k-\ell)\geq 0\), and let \(A_ x\) be the projection of \(A\) onto the plane \(y=0\). If for every compact set \(K\subset\Omega\), the intersection \(K\cap A_ x\) has zero capacity in \(\mathbb{R}^ m\) with respect to the potential \(| x|^{-\gamma}\) (or \(-\ln| x|\) if \(\gamma=0\)), then \(Pu=0\) in all of \(\Omega\). Theorem 2: Suppose \(m\) is even, and let \(L=L_ m(x,\partial_ x)+ L_ 1(x,y, \partial_ x, \partial_ y)\) have smooth coefficients, where \(L_ m(0, \partial_ x)\) is a homogeneous operator of order \(m\), elliptic with respect to \(x\), and \(\deg_ x L_ 1\leq m-1\). Suppose \(Lu=0\) in \(\Omega\setminus \{x=0\}\) and \(| u(x,y)|=o(\ln| x|)\) locally uniformly in \((x,y)\) as \(| x|\to 0\). If \(\int_{|\xi|=1} (L_ m(0,\xi))^{-1} d\sigma_ \xi \neq 0\), then \(Lu=0\) in all of \(\Omega\).
reviewed by: Harold P. Boas
Mathematics Subject Classification ID: 35B60
Mathematics Subject Classification ID: 35A20
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 130698
zbMATH Keywords: capacity
zbMATH Keywords: elliptic operators
zbMATH Keywords: removable singularities
zbMATH Keywords: linear partial differential equations
zbMATH Keywords: Laplacian
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 130698