Topology of rearrangements of non-smoothness sets of minimum functions in variational problems
type: Item
instance of: scholarly article
title: Topology of rearrangements of non-smoothness sets of minimum functions in variational problems
zbMATH Open document ID: 0722.49019
DOI: 10.1007/BF01077963
author: D. Kharzeev
publication date: 1990-01-01T00:00:00Z
review text: The minimal value of a smooth functional does not necessarily depend on the final point in a smooth manner. The non-smoothness set of this minimum function, called in optimal control the Bellman function, describes boundary conditions with a non-unique solution. In physics it corresponds to a shock wave. In time, this set may rearrange itself. \textit{I. A. Bogaevskij} [Leningr. Math. J. 1, No.4, 807-823 (1990); translation from Algebra Anal. 1, No.4, 1-16 (1989)] described typical rearrangements of the non-smoothness set of the minimum function in dimensions 3 and 4. However, it was observed [see \textit{S. N. Gurbatov} and \textit{A. I. Saichev}, Radiophys. Quant. Electron. 27, No.4, 303-313 (1984); translation from Izv. Vyssh. Uchebn. Zaved Radiofiz. 27, No.4, 456-458 (1984); and the joint work with \textit{S. F. Chandarin}, Mon. Not. R. Astron. Soc. 236, No.2, 385-402 (1989; Zbl 0662.76062)] that not all these rearrangements are realized as rearrangements of minimum functions of typical variational problems. For example, a spherical shock wave cannot be generated ``out of nothing''. Bogaevskij exhibited a list of rearrangements of non-smoothness sets of minimum functions of typical variational problems in dimensions \(\leq 4\). From this list it follows that the realizable rearrangements are distinguished by the following condition on the topology of the non- smoothness set: after a rearrangement, the non-smoothness set is locally contractible. In the present paper we describe another restriction on the type of rearrangement of the non-smoothness set of a minimum function of a variational problem, which consists of local coincidence of homotopy types of the complements of the non-smoothness sets at the moment of rearrangement and immediately following it. This restriction holds in arbitrary dimension, and not only for typical variational problems; it holds for all other ones as well, with the exception of a set of infinite codimension in a functional space. In l…
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 76L05
zbMATH DE Number: 4190531
zbMATH Keywords: Bellman function
zbMATH Keywords: shock wave
zbMATH Keywords: non-smoothness sets of minimum functions
MaRDI profile type: Publication
cites work: The large-scale structure of the Universe in the frame of the model equation of non-linear diffusion
full work available at URL: https://doi.org/10.1007/bf01077963
OpenAlex ID: W2047234698
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 4190531