14 - The Semismooth MPT

Summary

Soon after Baire had introduced in 1908 the concept of semicontinuity for real valued functions, Tonelli, in 1914, recognized semicontinuity as one of the relevant properties of the functionals of the calculus of variations.

L. Cesari, Optimization. Theory and applications, Springer-Verlag, 1983

A notion of critical points for continuous convex perturbations of C¹-functionals defined on a Banach space X is introduced, and an appropriate version of the MPT is proved. The same minimax argument serving to obtain a critical level in the classical version is used here too, to get a critical value corresponding to the new notion of critical point.

The main result of this chapter is taken from Szulkin [891]. It extends critical point theory to functionals of the form Φ + Ψ in a real Banach space X, where Φ: X → ℝ is a C¹-functional, and Ψ: X → ℝ ∪ {+∞} is a convex and lower semicontinuous (l.s.c.) perturbation. Denote by D(Ψ) = {u ∊ X; Ψ (x) <+∞} the effective domain of Ψ. The most important point here is to find an appropriate definition of nonregularity/regularity in this particular context and the rest of the program of projecting C¹-critical point theory for functionals of this form should be a priori easy.