The length of level lines of solutions of elliptic equations in the plane
type: Item
instance of: scholarly article
title: The length of level lines of solutions of elliptic equations in the plane
zbMATH Open document ID: 0664.35016
DOI: 10.1007/BF00251498
author: Giovanni Alessandrini
published in: Archive for Rational Mechanics and Analysis
publication date: 1988-01-01T00:00:00Z
review text: The author gives estimates for the length of \(x\in G:\quad u(x)=t\}\) where t is real number, G a subregion of a two-dimensional domain \(\Omega\) and u: \(\Omega\) \(\to {\mathbb{R}}\) denotes a nontrivial solution of the uniformly elliptic equation \[ 0=a_{11} u_{x_ 1x_ 1}+2a_{12} u_{x_ 1x_ 2}+a_{22} u_{x_ 2x_ 2}+b_ 1 u_{x_ 1}+b_ 2 u_{x_ 2} \] with sufficiently smooth coefficients. The estimate for the length of \(\{x\in G:\quad u(x)=t\}\) involves gradient bounds on subdomains \(G_ 1\) such that \(G\subset \subset G_ 1\subset \subset \Omega\) and is based on the fact that the critical points of u form a discrete subset of \(\Omega\) which in turn implies an integration by parts formula for the divergence of the field \(Du/| Du|\) on domains \(\{x\in G:\quad u(x)<t\}.\)
Mathematics Subject Classification ID: 35J15
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 4086147
zbMATH Keywords: uniformly elliptic
zbMATH Keywords: smooth coefficients
zbMATH Keywords: estimate
zbMATH Keywords: gradient bounds
zbMATH Keywords: critical points
zbMATH Keywords: integration by parts
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4086147