Minimization of nonsmooth integral functionals

Abstract

In this paper we examine optimization problems involving multidimensional nonsmooth integral functionals defined on Sobolev spaces. We obtain necessary and sufficient conditions for optimality in convex, finite dimensional problems using techniques from convex analysis and in nonconvex, finite dimensional problems, using the subdifferential of Clarke. We also consider problems with infinite dimensional state space and we finally present two examples.

Keywords

• Sobolev space
• convex integrand
• convex subdifferential
• Clarke's subdifferential
• ϵ-subdifferential
• Young-Fenchel equality.