A uniqueness theorem for \(Z^ n\)-periodic variational problems
type: Item
instance of: scholarly article
title: A uniqueness theorem for \(Z^ n\)-periodic variational problems
zbMATH Open document ID: 0634.49018
DOI: 10.1007/BF02564459
author: Victor Bangert
published in: Commentarii Mathematici Helvetici
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140095
review text: Let \(F: R^ n\times R\times R^ n\to R\) be an appropriate integrand which is Z-periodic in the first \(n+1\) variables. Moser's variational problem consists in finding functions \(u: R^ n\to R\) which minimize \(\int F(x,u,u_ x)dx\) with respect to all compactly supported variations of u. Minimal solutions without self-intersections of this problem have a uniquely determined ``slope'' \(\alpha \in R^ n\). For irrational \(\alpha\) these minimal solutions are obtained as limits of minimal solutions with rational slope. The author describes a new approach to the basic invariant, the slope of a minimal solution without selfintersections, and shows that for every \(\alpha\in R^ n\setminus Q\) n the limit set of minimal solutions is essentially unique.
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 49Q15
zbMATH DE Number: 4032482
zbMATH Keywords: Z-periodic
zbMATH Keywords: variational problem
zbMATH Keywords: Minimal solutions without self- intersections
zbMATH Keywords: slope
zbMATH Keywords: limit set
reviewed by: Constantin Udrişte
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4032482