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 ƒ, 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.