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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 $$\Omega' \subset \overline{\Omega'}\subset\Omega$$.
##### MSC:
 35F15 Boundary value problems for linear first-order PDEs 35B65 Smoothness and regularity of solutions to PDEs 47G30 Pseudodifferential operators