A note on quasi-periodic solutions of some elliptic systems
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 \[ \langle U \rangle \equiv \int_{T^N+M} U(\theta,\psi, \varepsilon) {d\theta d\psi \over (2\pi)^{N +M}} = 0. \tag{5} \] In fact (5) is a normalizing condition necessary to have uniqueness.
scientific article; zbMATH DE number 880626