Fredholm quasilinear manifolds and degree of a Fredholm quasilinear mapping between them
type: Item
instance of: scholarly article
title: Fredholm quasilinear manifolds and degree of a Fredholm quasilinear mapping between them
zbMATH Open document ID: 1263.47070
author: D. Kharzeev
published in: Ukrainian Mathematical Journal
publication date: 2012-03-20T00:00:00Z
full work available at URL: http://dspace.nbuv.gov.ua/handle/123456789/166034
review text: The author continues his study of quasilinear Fredholm mappings [the author, Turk. J. Math. 24, No. 1, 1--14 (2000; Zbl 0971.47045); ibid. 28, No. 3, 205--215 (2004; Zbl 1077.58003); ibid. 30, No. 2, 129--138 (2006; Zbl 1108.55004)] with the intention to generalize \textit{A. I. Shnirel'man}'s results [Mat. Sb., N. Ser. 89(131), 366--389 (1972; Zbl 0249.55003)]. Let \(M\) and \(N\) be compact smooth manifolds of dimension \(m<n\), where \(N\) is assumed to be closed. The author's aim is to prove that the Sobolev manifold \(H_s(M,N)\) is an FQL-manifold as defined by Šnirel'man [loc.\,cit.]. In order to achieve this, he views \(N\) as a submanifold of \(\mathbb{R}^{2n+1}\). Let then \(u\in H_s(M,\mathbb{R}^{2n+1})\) with \(\|u\|_s<j\). Alas, there are some puzzling facts in the proof: On p.\,677, line 5, the author estimates \(\|u\|_{C'}\) which is nowhere defined. Maybe it is \(\|u\|_{C^1}\) which also appears later, but would that not require a condition on \(s\)? On the following page, on line 1, he estimates \(\|u\|_{C^1}\leq K\|u\|_s\) ``for sufficiently large \(s\)'' -- but \(s\) is fixed and might not be ``sufficiently large''. Moreover, on the same page, line 4, he claims that (with \(K\) the constant from above and \(d\) the metric on \(M\)), for any \(u\in H_s(M,\mathbb{R}^{2n+1})\) with \(\|u\|_s<j\) and any \(x',x''\in M\), we have \(\|u\|_s<Kjd(x',x'')\), which is absurd even if one had \(\leq\) instead of \(<\). The author then proceeds to the case \(m\geq n\) and he finally tries to define a degree function for FSQL-mappings (cf.\ the first of the author's papers listed above for the definition) \(f:X\to X'\) with FQL-manifolds \(X\) and \(X'\) embedded in Banach spaces \(E_1\) and \(E_2\) satisfying the awkward hypothesis \(\|x\|_1\leq \|f(x)\|_2\) whenever \(x\in X\). This degree has the property that \(\deg(f, \cdot)\) is constant on \(X'\). As a matter of fact, the degree is defined only up to sign, so it is more or less a mod 2-degree. The interes…
Mathematics Subject Classification ID: 47H11
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 55M25
Mathematics Subject Classification ID: 58B15
zbMATH DE Number: 6016083
zbMATH Keywords: Fredholm special quasi-linear manifold
zbMATH Keywords: Fredholm special quasi-linear map
zbMATH Keywords: degree
reviewed by: Christian Fenske
MaRDI profile type: Publication
OpenAlex ID: W2012725407
cites work: A special quasi-linear mapping and its degree
cites work: Q4673769
cites work: L-homology theory of FSQL-manifolds and the degree of FSQL-mappings
cites work: NON-LINEAR FREDHOLM MAPS AND THE LERAY-SCHAUDER THEORY
cites work: Q5636087
cites work: Q3990190
DOI: 10.1007/S11253-011-0534-8
description: scientific article; zbMATH DE number 6016083