40A05-InfiniteDimensionalHamiltonianSystem1.tex

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An infinite dimensional Hamiltonian system takes the form
    $$ 
       (IHS) \;\; \left \{ \begin{array}{rll}
               \partial_t u -\Delta_x u &= H_v(t,x,u,v) & \\
               -\partial_t v -\Delta_x v &= H_u(t,x,u,v)  & 
\forall (t,x) \in \mathbb{R} \times \Omega
         \end{array}  \right.
   $$
where $\Omega\subset {\mathbb R}^N$ is a smoothly bounded domain,
$H \in \mathcal{C}^1 (\mathbb{R}\times \overline{\Omega} \times \mathbb{R}^2, \mathbb{R})$. 


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