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Convergence to steady states or periodic solutions in a class of Hamilton-Jacobi equations. (English) Zbl 0979.35033

In this paper are studied the viscosity solutions of two classes of Hamilton-Jacobi equations, namely \[ \begin{cases} u_t+H(x,Du)=0,\;x\in{\mathcal M},\;\text{resp. }x\in \Omega,\\ u(t,x)=\varphi(x)\text{ on }\partial\Omega,\end{cases} \tag{1} \] and the time-periodic version of (1) \[ \begin{cases} u_t+H(t,x,Du)=0,\;x\in{\mathcal M},\;\text{resp. }x\in \Omega,\\ u(t,x)=\varphi(t,x)\text{ on }\mathbb{R}\times \partial\Omega,\end{cases} \tag{2} \] where \(\mathcal M\) is a smooth compact \(N\)-dimensional manifold without boundary, and \(\Omega\) is an open bounded subset of \(\mathbb{R}^N\).
The purpose of this work is to give a different proof of a convergence theorem – obtained by A. Fathi – in the autonomous case (1) when the equations are posed on a manifold, and to extend it to Dirichlet boundary conditions on an open subset.
When the equations are time-periodic, the convergence in several nontrivial special cases is proved.

MSC:

35F30 Boundary value problems for nonlinear first-order PDEs
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
35B10 Periodic solutions to PDEs
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