On the Lavrentiev phenomenon for autonomous second-order integrands
type: Item
instance of: scholarly article
title: On the Lavrentiev phenomenon for autonomous second-order integrands
zbMATH Open document ID: 0812.49011
DOI: 10.1007/BF00375694
author: Chih-Wen Cheng
author: Victor J. Mizel
published in: Archive for Rational Mechanics and Analysis
publication date: 1995-05-07T00:00:00Z
review text: The so-called Lavrentiev phenomenon is related to the question of whether the infimum of a functional on the class of absolutely continuous functions is strictly lower than the infimum of the same functional on a dense subspace of this class. This phenomenon was first introduced by M. A. Lavrentiev in 1926. The article clarifies the result presented by \textit{F. H. Clarke} and \textit{R. B. Vinter} [Trans. Am. Math. Soc. 291, 73-98 (1985; Zbl 0563.49009)] ensuring the non-existence of a Lavrentiev gap in the case of autonomous first-order integrands on \(\mathbb{R}^ n\)-valued \(u\). The authors present theorems and related examples showing that for certain autonomous second- order integrands a Lavrentiev gap occurs for a dense subclass \(W^{2,p} (0,1)\) of the function space \(W^{2,1}(0,1)\) if one considers nonnegative functions \((u\geq 0)\). On the other hand, the gap typically fails to take place for smooth integrands over the full function space (i.e, if one allows functions possessing fixed sign). By considering additive perturbations of the original problem, the authors obtain a class of fully regular problems exhibiting the Lavrentiev gap.
Mathematics Subject Classification ID: 49J35
zbMATH DE Number: 635832
zbMATH Keywords: minimax problems
zbMATH Keywords: Lavrentiev phenomenon
zbMATH Keywords: autonomous second-order integrands
zbMATH Keywords: Lavrentiev gap
reviewed by: I. N. Lushchakova
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 635832