Weak Lower Semicontinuity of a Functional with Any Order

On a Restricted Weak Lower Semicontinuity for Smooth Functional on Sobolev Spaces

We study a restricted weak lower semicontinuity property, which we call the (PS)-weak lower semicontinuity, for a smooth integral functional on the Sobolev space along all weakly convergent Palais-Smale sequences of the functional. By the Ekeland variational principle, the (PS)-weak lower semicontinuity is sufficient for the existence of minimizers under the usual coercivity assumption. In gene...

Weak Lower Semicontinuity for Polyconvex Integrals in the Limit Case

We prove a lower semicontinuity result for polyconvex functionals of the Calculus of Variations along sequences of maps u : Ω ⊂ R → R in W , 2 ≤ m ≤ n, bounded in W 1,m−1 and convergent in L under mild technical conditions but without any extra coercivity assumption on the integrand.

On the role of lower bounds in characterizations of weak lower semicontinuity of multiple integrals

We study weak lower semicontinuity of integral functionals in W 1,p under standard p-growth conditions, with integrands whose negative part may have p-growth as well. MSC 2000: 49J45

A - Quasiconvexity , Lower Semicontinuity And

Lower Semicontinuity of the Willmore Functional for Currents

The weak mean curvature is lower semicontinuous under weak convergence of varifolds that is if μk → μ weakly as varifolds then ‖ ~ Hμ ‖Lp(μ)≤ lim infk→∞ ‖ ~ Hμk ‖Lp(μk) . In contrast, if Tk → T weakly as integral currents then μT may not have locally bounded first variation even if ‖ ~ HμTk ‖L∞(μk) is bounded. In 1999, Luigi Ambrosio asked the question whether lower semicontinuity of the weak m...