A note on localized directional weak lower semi-continuity
Erwin Brüning · Quaestiones Mathematicae · 2006
In minimization problems for functionals f : M → R, M ⊆ E a subset of some infinite dimensional Banach space E, we typically have to rely on weak (sequential) lower semi-continuity of f on the whole space E even if M is a proper subset of E. The main reason for this lack of 'localized' weak lower semi-continuity seems to be that it is not known how to get and/or to characterize weak sequential lower semi-continuity on a subset M without knowing it on the whole space. As a first step to overcome this difficulty we propose the concept of 'localized directional weak sequential lower semi-continuity' and offer a way to implement it, namely in terms of conditions on the Gateaux derivative f′ of f (weak K-monotonicity). This allows to formulate a criterium and new sufficient conditions for the existence of a minimizer. We conclude with a discussion of applications to the variational approach to the solution of (systems of) nonlinear partial differential equations where we focus on the case of integral functionals of vector fields for which the integrand is not assumed to be quasi-convex.