Exotic nonlinear analysis. III: Yang-Mills quantum fields
type: Item
instance of: scholarly article
title: Exotic nonlinear analysis. III: Yang-Mills quantum fields
zbMATH Open document ID: 1153.47050
author: Jean-Bernard Pécot
published in: Dynamics of Partial Differential Equations
publication date: 2008-04-04T00:00:00Z
review text: The article presents the following inverse function theorem: (1) Consider a sequence of Banach spaces \((E_n)_{n\geq 0}\) with continuous inclusions: \[ \dots\subset E_{n+1} \subset E_n \subset E_{n-1} \subset \dots \subset E_0 \] with \[ \forall n \geq 0, \quad \|x\|_{E_{n+1}} \geq \|x\|_{E_n}. \] (2) \(f\) is an operator in \(C^0(E_{n+2},{\mathcal F}_n)\), for any \(n \geq 0\), where \(({\mathcal F})_{n\geq0}\) is a sequence of Banach spaces with continuous inclusions: \[ \dots\subset {\mathcal F}_{n+1} \subset {\mathcal F}_n \subset {\mathcal F}_{n-1} \subset \dots \subset {\mathcal F}_0, \] and for any \(x,y\in E_{n+2}\), the difference \(f(x)-f(y)\in F_n\), where \((F_n)_{n\geq0}\) is another sequence of Banach spaces with \[ F_n \subset {\mathcal F}_n, \quad n \geq 0 \;(\text{with a continuous injection}), \qquad \|\cdot\|_{{\mathcal F}_n} = \|\cdot\|_{F_n}. \] Suppose that \(Df(x)\) for any \(x \in E_{n+2}\), \(n \geq 0\), exists in \(L(E_{n+2},F_n)\) and that \(x \mapsto Df(x)\) is in \(C^0(E_{n+2},L(E_{n+2},F_n))\). (3) Consider a sequence \((A_n)_{n\geq 0}\) of sets with: (a) \(\forall n \geq 0\), \(m \geq 0\): \(A_n \subset E_m\); (b) \(\forall a_n \in A_n\), \(\forall m \geq 0\): \(\|a_n\|_{E_m} \leq \|a_n\|_{E_n}\). (4) For any \(a_m \in A_m\), \([Df(a_m)]^{-1}\) exists in \(L(E_n,E_{n+1})\) for any \(n \geq 0\), with \[ \|[Df(a_m)]^{-1}\|_{L(F_n,E_{n+1})} \leq \alpha_m \|a_m\|_{E_m}^r + \beta_m, \] for a real \(r\) independent of \(m\) and of \(n\), where \(\alpha_m\) and \(\beta_m\) are positive numbers only depending on \(m\) and so independent of \(n\). (5) There exists a positive real \(\gamma\) independent of \(n\) such that for any couple \((x,y)\) in \(E_{n+2} \times E_{n+2}\): \[ \|Df(x) - Df(y)\|_{L(E_{n+2},F_n)} \leq \gamma(\|x\|_{E_{n+2}} + \|y\|_{E_{n+2}} + 1)\|x - y\|_{E_{n+2}}. \] (6) For any \(m \geq 0\): \(12\beta_m\gamma > 1\). (7) For any \(n \geq 0\) and \((x,y) \in E_{n+1}^2\): \[ \|x - y\|_{E_{n+1}} \leq c_n(x,y)\|f(x) - f(y)\|_{F_…
Mathematics Subject Classification ID: 47J07
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 81T13
zbMATH DE Number: 5259046
zbMATH Keywords: inverse function theorem
zbMATH Keywords: Yang-Mills gauge fields
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
DOI: 10.4310/DPDE.2007.V4.N4.A2
description: scientific article