# zbMATH — the first resource for mathematics

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 $$H(x, \nabla u)+ d- f(x)\leq \varepsilon$$.

##### MSC:
 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)
##### Keywords:
Hamilton-Jacobi equations