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A regularity result for Hamiltonian systems. (English) Zbl 1006.35021
The authors consider the Hamiltonian system \[ \begin{aligned} \dot x_j & =-(\partial_{\xi_j}p) (t,x,\xi),\\ \dot\xi_j & =-(\partial_{x_j}p) (t,x,\xi),\\ x(0) & = y,\\ \xi(0) & =\eta,\end{aligned} \tag{1} \] where \(p\) is a \(\mathbb{C}^\infty\) function of the variables \((t,x,\xi)\in(-T,T) \times\Omega \times\mathbb{R}^n\), \(\Omega \subseteq\mathbb{R}^n\) open, satisfying the estimates \[ \bigl|\partial_x^\alpha \partial^\beta_\xi \partial^\gamma_t p(t,x,\xi) \bigr|\leq C_{\alpha \beta\gamma K} \psi(\xi)^{1-|\beta|} \tag{2} \] with \(K\subset \Omega\) compact and \(\psi\) an appropriate weight function. This Hamiltonian system appears in the study of the boundary value problem \[ \partial_tu= p(t, x,\partial_x), \quad u|_{t=0}=u_0\in {\mathcal E}'(\Omega) \] where \({\mathcal E}' (\Omega)\) denotes the space of distributions with compact support, when a solution of the form \[ u(t,x)=\int_{\mathbb{R}^n} e^{i\varphi(t,x,\eta)} \lambda (t,x,\eta)\widehat u_0(\eta) d\eta \] is proposed for a phase \(\varphi\) and an amplitude \(\lambda\) to be found satisfying estimates (2). The authors show by means of an iterative argument that the system (1) has a solution \((x(t,y, \eta)\), \(\xi(t,y,\eta))\) that is a \(C^\infty\) function on \((-T,T)\times \Omega' \times\mathbb{R}^n\), where \(\Omega'\) is an open subset of \(\Omega\) satisfying \(\Omega' \subset \overline{\Omega'}\subset\Omega\).
35F15 Boundary value problems for linear first-order PDEs
35B65 Smoothness and regularity of solutions to PDEs
47G30 Pseudodifferential operators