A note on quasi-periodic solutions of some elliptic systems
type: Item
instance of: scholarly article
title: A note on quasi-periodic solutions of some elliptic systems
zbMATH Open document ID: 0860.35027
DOI: 10.1007/BF00916825
author: L. Chierchia
author: Corrado Falcolini
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 1997-04-29T00:00:00Z
review text: The paper is concerned with the construction of quasi-periodic solutions of systems of nonlinear partial differential equations, for the unknown vector function \(u:y \in \mathbb{R}^M \to u(y) \in\mathbb{R}^N\), of the type: \[ \Delta u_i = \varepsilon {\partial f\over \partial x_i} (u, y), \quad i=1,2, \dots, N, \tag{1} \] where \((x,y)\in \mathbb{R}^{N+M}\), \(\Delta= \sum^M_{j=1} (\partial^2/ \partial^2_j)\), \(f(x,y)\) is smooth and periodic in each of its \(N+M\) variables and \(\varepsilon\) is a parameter. Here ``quasi-periodic'' means that there exists an \(N\times M\) matrix \(\Omega\), and a smooth periodic function \(U: (\theta, \psi) \in T^{N+M} \equiv \mathbb{R}^{N +M}/2 \pi \mathbb{Z}^{N+M} \to U(\theta, \psi) \in\mathbb{R}^N\), such that (2) \(u(y)= \Omega y+ U(\Omega y,y)\), \(\forall y\in \mathbb{R}^M\). The existence of ``quasi-periodic'' solutions is motivated by the so-called Kolmogorov-Arnold-Moser theory for variational problems. In this note quasi-periodic solutions are constructed adapting the so-called Siegel's direct method. The essential assumption is the following ``Diophantine'' condition: Assume that the \(M\times (N+M)\) matrix \(A= (\Omega^T,I_M)\) satisfies \[ |Al |\geq {1\over \gamma |l|^\tau} \quad\forall l \in \mathbb{Z}^{N+M} \backslash \{0\}. \tag{3} \] It is a straightforward check that \(u\) from (2) satisfies (1) if and only if \(U(\theta, \psi)\) solves the degenerate system on \(T^N+M\) given by \[ LU = \varepsilon f_x (\theta+U, \psi), \quad L \equiv \sum^M_{j=1} (\omega_j \cdot \partial_\theta + \partial_{\psi_j})^2. \tag{4} \] Consequently the following result is proved: Let \(f\) in (1) be real-analytic on \(T^{N+M}\) and let \(\Omega\) satisfy (3). Then there exists a unique solution \(U=U (\theta, \psi, \varepsilon)\) of (4) jointly real-analytic in \((\theta,\psi)\) and \(\varepsilon\) (for \(\varepsilon\) in a complex neighbourhood of \(\varepsilon=0)\), such that \[ \langle U \rangle \equiv \int_{T^N+M} U(\theta,\p…
reviewed by: Ivan Ginchev
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 35J45
zbMATH DE Number: 880626
zbMATH Keywords: existence
zbMATH Keywords: semilinear elliptic systems
zbMATH Keywords: degenerate system
zbMATH Keywords: normalizing condition
zbMATH Keywords: uniqueness
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00916825
OpenAlex ID: W1977553310
description: scientific article; zbMATH DE number 880626