A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions

We prove that if X is a Banach space which admits a smooth Lipschitzian bump function, then for every lower semicontinuous bounded below function f(hook), there exists a Lipschitzian smooth function g on X such that f + g attains its strong minimum on X, thus extending a result of Borwein and Preiss. We then show how the above result can be used to obtain existence and uniqueness results of viscosity solutions of Hamilton-Jacobi equations in infinite dimensional Banach spaces a without assuming the Radon Nikodym property.