Regularity of solutions in the fundamental problem of the classical calculus of variations
type: Item
instance of: scholarly article
title: Regularity of solutions in the fundamental problem of the classical calculus of variations
zbMATH Open document ID: 0807.49033
DOI: 10.1007/BF01236626
author: Felix Sadyrbaev
published in: Mathematical Notes
publication date: 1994-07-21T00:00:00Z
review text: This note considers the fundamental problem of the classical calculus of variations, which means the problem of minimizing the functional \(\int^ b_ a L(t,x(t),x'(t))dt\) in the class of absolutely continuous functions \(x: [a,b]\to\mathbb{R}^ n\) satisfying the boundary conditions \(x(a)= A\) and \(x(b)= B\) with \(A,B\in \mathbb{R}^ n\). Assume that the Lagrangian \(L(t,x,y)\) is of \(C^ 1\) and strictly convex with respect to \(y\). The author proves that if for \(| y|> \lambda\) sufficiently large, 1. \(L(t,x,y)> \Lambda(b- a)^{-1}\), where \(\Lambda\) is the value of the problem; 2. \(L^ 2_ y(t,x,y)> N> p= \max_{| y|\leq \lambda} L^ 2_ y(t,x,y)\); 3. \(|(L_ x,L_ y)|< K\varphi(L^ 2_ y)L\) with \(\int^ N_ p ds/\varphi(s)> 2K\Lambda\); then every solution of the problem is smooth.
Mathematics Subject Classification ID: 49N60
zbMATH DE Number: 578696
zbMATH Keywords: regularity
zbMATH Keywords: fundamental problem
zbMATH Keywords: calculus of variations
zbMATH Keywords: Lagrangian
MaRDI profile type: Publication
cites work: One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation
cites work: Q3995739
cites work: Existence and regularity in the small in the calculus of variations
cites work: Q3349483
Wikidata QID: Q57554724
description: scientific article; zbMATH DE number 578696