Volume 5, Issue 2

Abstract. We give a new proof of sequential weak* lower semicontinuity in for integral functionals of the form where and is a quasiconvex Carathéodory integrand with linear growth at infinity, i.e. for some , and such that the recession function exists and is (jointly) continuous. In contrast to the classical proofs by Ambrosio and Dal Maso [ J. Funct. Anal. 109 (1992), 76–97] and Fonseca and Müller [ Arch. Ration. Mech. Anal. 123 (1993), 1–49], we do not use Alberti's Rank-One Theorem [ Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), 239–274], but a rigidity result for gradients. The proof is set in the framework of generalized Young measures and proceeds via establishing Jensen-type inequalities for regular and singular points of .

Abstract. We deal with mappings defined between Riemannian manifolds that belong to trace spaces of Sobolev functions, see , , , , , . Such mappings are equipped with a natural energy, equivalent to the fractional norm. We study the class of Cartesian currents that arise as weak limits of sequences of mappings with equibounded energies. Under suitable topological assumptions on the domain and target manifolds, we prove a density property of graphs of smooth maps. As a consequence, we discuss the corresponding relaxed energy. For mappings with values into the sphere, an explicit formula for the relaxed energy is obtained.

Abstract. Isoperimetric problems as well as free minimum problems for non-coercive, one-dimensional functionals are investigated. Models are the classical problem of the catenary, the brachistochrone, and Newton's problem of minimal resistance. A convex minimizer is shown to exist in classes of competing functions containing both convex and non-convex elements. All problems are solved by means of convex rearrangement , a transformation that turns an absolutely continuous function into a convex one having same boundary values and same graph length but less energy.