Existence and regularity of extrema

Abstract

Consider the minimization of a possibly noncoercive Gâteaux differentiable functional F:X→ R . A modified notion of coercivity is introduced which may be usable to show existence of a minimum. Alternatively, F ̂ :D→ R has a minimum at yε D ( F ̂ not differentiable but the restriction F of F ̂ to X ⊂ D differentiable), one may be able to show y ̄ is actually in X . The latter case is related to justification of formally calculated “necessary conditions” for optimal controls. The arguments are applications of Ekeland's “approximate variational principle” ( J. Math. Anal. Appl. 47 (1974) , 324–353).