Classical global solutions for a class of Hamilton-Jacobi equations
type: Item
instance of: scholarly article
title: Classical global solutions for a class of Hamilton-Jacobi equations
zbMATH Open document ID: 1207.35093
author: Jaime Cruz-Sampedro
published in: Qualitative Theory of Dynamical Systems
publication date: 2010-10-04T00:00:00Z
review text: Let \(V\) be a real-valued \(C^2\)-function on \(\mathbb R^n\), \(n\geq 2\), such that \(V(x)=0\) for \(|x|\leq R\) and \(\partial_x^\alpha V(x)=O(|x|^{-\epsilon-|\alpha|})\) (as \(|x|\to\infty\)) for some \(\epsilon>0\) and all multiindices with \(|\alpha|\leq 2\). Using the method of characteristics and a global inverse function theorem of Hadamard the authors shows that, for sufficiently large values of \(R>0\), the Hamilton-Jacobi equation of eikonal type, \[ |\nabla u(x)|^2+V(x) = k^2,\quad k>0, \] has a \(C^1\)-solution on \(\mathbb R^n\setminus\{0\}\).
Mathematics Subject Classification ID: 35F05
Mathematics Subject Classification ID: 35C05
zbMATH DE Number: 5795531
zbMATH Keywords: Hamilton-Jacobi equation
zbMATH Keywords: eikonal equation
zbMATH Keywords: global solution
reviewed by: R. L. Schilling
MaRDI profile type: Publication
full work available at URL: s12346-010-0017-6
OpenAlex ID: W1980469664
cites work: Q4730138
cites work: On eikonal equations associated with schrodinger operators with nonspherical radiation conditions
cites work: Exact asymptotic behavior at infinity of solutions to abstract second-order differential inequalities in Hilbert spaces
cites work: Q5185568
cites work: Long-range scattering in the position representation
cites work: Das asymptotische Verhalten von Lösungen eines Typs von Differentialgleichungen
cites work: Partial differential equations. 4th ed
cites work: Wave operators for the Schrödinger equation
DOI: 10.1007/S12346-010-0017-6
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description: scientific article; zbMATH DE number 5795531