Exotic nonlinear analysis. III: Yang-Mills quantum fields
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 \((x,y) \in E_{n+1}^2\): \[ \|x - y\|_{E_{n+1}} \leq c_n(x,y)\|f(x) - f(y)\|_{F_n}, \] where \(c_n\) is a continuous operator from \(E_{n+1} \times E_{n+1}\) into \({\mathbb R}^+\). (8) \(x \mapsto f(x) - f(0)\) acts from \(A_m\) into \(B_{\alpha(m)}\), where \(\alpha\) is a function from \({\mathbb N}\) into \({\mathbb N}\) and \((B_m)_{m\geq 0}\) a sequence of sets such that: (a) \(\forall n, m \geq 0\): \(B_n \subset F_m\). (b) \(\forall b_n \in B_n\), \(\forall m\), \(m \geq 0\): \(\|b_n\|_{F_m} \leq \|b_n\|_{F_n}\). (9) We consider a sequence of sets \((C_n)_{n\geq 0}\) with: (a) \(\forall n, m \geq 0\): \ \(C_n \subset F_m\) (b) \(\forall c_n \in C_n\), \(\forall m \geq 0\): \(\|c_n\|_{F_m} \leq \|c_n\|_{F_n}\), (c) \(\bigcup\limits_{n\geq0} C_n\) is dense in \(F_m\), for any \(m \geq 0\). Then, for any \(a_m\) in \(A_m\) (this for any \(m \geq 0\)) and for any \(n \geq 0\), \(f^{-1}\) exists as a function from \(B_{F_n}\left(f(a_m),\frac{r_m}{\alpha_m \|a_m\|_{E_m}^r + \beta_m}\right)\) into \(B_{E_{n+1}}(a_m,2r_m)\), for any \(r_m\) such that \[ r_m<\frac1{4(\alpha_m\|a_m\|_{E_m}^r+\beta_m)\times\gamma(2\|a_m\|_{E_m}+3)}. \] The analysis of the assumptions and the statement in this theorem is absent. Based on this theorem, the author proves the existence of an infinity of smooth classical symmetric Yang--Mills gauge fields in a smooth \(3\)-dimensional compact manifold with boundary for any finite interval of time or in whole \({\mathbb R}^4\). At the end of the article, it is stated that the theorem above is applied in some questions of existence and uniqueness for nonlinear hyperbolic and parabolic differential equations. [Editor's remark: Parts I and II are not referenced in the article as separate publications. It appears from the author's Introduction that Parts I, II, III correspond to Sections 2, 3, 4 of the present paper.]
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