On points of qualitative semicontinuity

On the Algebraic Representation of Semicontinuity

Abstract: The concepts of upper and lower semicontinuity in pointfree topology were introduced and first studied by Li and Wang in 1997. However Li and Wang’s treatment does not faithfully reflect the original classical notion. In this note, we present algebraic descriptions of upper and lower semicontinuous real functions, in terms of frame homomorphisms, that suggest the right alternative to ...

On the Lower Semicontinuity of Quasiconvex Integrals

In this paper, dedicated to Neil Trudinger in the occasion of his 70 birthday, we propose a relatively elementary proof of the weak lower semicontinuity in W 1,p of a general integral of the Calculus of Variations of the type (1.1) below with a quasiconvex density function satisfying p-growth conditions. Several comments and references on the related literature, and a subsection devoted to some...

On L-lower semicontinuity in BV

and of the corresponding extension (1.3) to BV (Ω), with the aim of understanding which are the minimal assumptions on f that guarantee lower semicontinuity. Here, the starting point is the result, proved in [20], stating that if the integrand f(x, u, ξ) is a nonnegative, continuous function from Ω× IR× IR , convex in ξ, and such that the (classical) derivatives ∇xf,∇ξf,∇x∇ξf exist and are cont...

Semicontinuity of Convex-valued Multifunctions

We introduce semicontinuity concepts for functions f with values in the space C(Y ) of closed convex subsets of a finite dimensional normed vector space Y by appropriate notions of upper and lower limits. We characterize the upper semicontinuity of f : X → C(Y ) by the upper semicontinuity of the scalarizations σf( · )(y∗) : X → R by the support function. Furthermore, we compare our semicontinu...

Lower semicontinuity of surface energies