Direct methods in variational field theory
type: Item
instance of: scholarly article
title: Direct methods in variational field theory
zbMATH Open document ID: 1502.49002
DOI: 10.1134/S0037446622050056
author: Richard Gratwick
author: M. A. Sychev
published in: Siberian Mathematical Journal
publication date: 2022-10-06T00:00:00Z
review text: In this short paper, the authors show that the classical Weierstras-Hilbert field theory can be extended. From the fact that for each extremal field it is true that if an extremal is an element of the field, then a minimum is attained in the class of Sobolev functions with the same boundary data as for the extremal and with graphs in the set covered by the field, the authors derive that this result remains valid if one of the extremals is singular. On the other hand, it is shown that if there is a field containing more than one singular extremal, then each of these extremals defines the minimization problem with having no solution in the class of Lipschitz functions with graphs in the set covered by the field. The new results are collected in the first section and the rest of the paper is dedicated to the proofs of the main theorem, what makes it to an easy to read contribution to this field.
reviewed by: Uwe Prüfert
Mathematics Subject Classification ID: 49J21
Mathematics Subject Classification ID: 49K05
zbMATH DE Number: 7596647
zbMATH Keywords: integral functional
zbMATH Keywords: ellipticity
zbMATH Keywords: Euler equation
zbMATH Keywords: minimizer
zbMATH Keywords: Weierstras-Hilbert field theory
zbMATH Keywords: direct method
zbMATH Keywords: singular extremal
MaRDI profile type: Publication
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full work available at URL: s0037446622050056
OpenAlex ID: W4300594497
description: scientific article; zbMATH DE number 7596647