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Some ergodic problems for Hamilton-Jacobi equations in Hilbert space. (English) Zbl 0848.35026
Existence and uniqueness results for viscosity solutions of Hamilton-Jacobi equations of the type \(H(x, \nabla u_\lambda(x))+ \lambda u_x(x)- f(x)= 0\) in \(\Omega\) with Neumann boundary conditions, where \(\Omega\) is a domain in a Hilbert space are established using Perron’s method. The limit of \(\lambda u_\lambda(x)\) as \(\lambda\to \infty\) is the same constant \(d\) for each \(x\). The constant \(d\) is characterized through viscosity solutions of \(H(x, \nabla u)+ d- f(x)\leq \varepsilon\).

35F30 Boundary value problems for nonlinear first-order PDEs
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables)