Another theorem of classical solvability `in small' for one-dimensional variational problems
type: Item
instance of: scholarly article
title: Another theorem of classical solvability `in small' for one-dimensional variational problems
zbMATH Open document ID: 1257.49039
published in: Archive for Rational Mechanics and Analysis
publication date: 2012-11-05T00:00:00Z
review text: A direct method for studying local minimizers of the variation problem \[ \int_a^bL(x,u(x),\dot{u}(x))dx\rightarrow \min,\;\; L(x,u,v):[a,b]\times R^{2}\rightarrow R, \] \[ u(a)=A,u(b)=B,\;\; u:[a,b]\rightarrow R \] is suggested on the basis of a regularity theory (a-priori estimates and compactness in \(C^1\)). The following aspects are investigated: 1. Is each weak local minimizer also a strong local minimizer? 2. Let \(u(\cdot)\) be a local minimizer and \(L\in C^{\infty},L_{vv}>0.\) Let \(L_n\in C^{\infty},(L_n)_{vv}>0\) and let \(\|L_n-L\|\rightarrow 0\) as \(n\rightarrow \infty .\) Is it correct that there exist local minimizers \(u_n(\cdot)\) associated with \(L_n\) such that \(\|u_n-u\|\rightarrow 0?\) 3. What are necessary and sufficient conditions to admit a local minimizer?
reviewed by: T. Tadumadze
Mathematics Subject Classification ID: 49N60
Mathematics Subject Classification ID: 49J99
Mathematics Subject Classification ID: 49K99
zbMATH DE Number: 6101952
zbMATH Keywords: variational problem
zbMATH Keywords: existence of local minimizer
zbMATH Keywords: necessary and sufficient conditions for local minimizer
author: M. A. Sychev
MaRDI profile type: Publication
full work available at URL: s00205-011-0416-0
OpenAlex ID: W1975741129
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DOI: 10.1007/S00205-011-0416-0
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