Topology of Sobolev Mappings

Abstract

This paper addresses some topological and analytical issues concerning Sobolev mappings between compact Riemannian manifolds. Among the results we obtained are unified proofs of various generalizations of results obtained in a recent work of Brezis and Li. In particular we solved two conjectures in [BL]. We also give a topological obstruction for the weak sequential density of smooth maps in a given Sobolev mapping space. Finally we show a necessary and sufficient topological condition under which the smooth maps are strongly dense in the Sobolev spaces. The earlier result, Theorem 1 of [B2], was shown to be not correct.

Sobolev Mappings: Lipschitz Density is not a Bi-Lipschitz Invariant of the Target

“…Later, mainly due to the work of Bethuel, [5], it turned out that the condition for the density of smooth mappings can be formulated in terms of algebraic topology describing the topological structure of the manifolds M and N . Finally a necessary and sufficient condition for the density of smooth mappings in the space of Sobolev mappings W 1,p (M, N ) was discovered by Hang and Lin [29], [30]. This result corrects an earlier statement of Bethuel [5] about a necessary and sufficient condition for the density.…”

“…This result corrects an earlier statement of Bethuel [5] about a necessary and sufficient condition for the density. Other papers on this topic include [4], [6], [7], [9], [10], [11], [12] [15], [16], [19], [20], [23], [24], [28], [29], [30], [34], [35], [50], [58], [59].…”

Partial Regularity for Holonomic Minimisers of Quasiconvex Functionals

“…Comparison maps satisfying the non-linear manifold constraints in the indirect blow-up proof are obtained from a particular extension lemma of Luckhaus (cf. [63,64] and also [48,55,78]), however this construct only works in the blow-up case due to the order in which limits are taken (cf. [34, p. 87]).…”

Mathematics in the age of the Turing machine

“…The corrected theorem appears in [HL03]. Such examples are so plentiful that a Wiki page has been set up to classify them, with references to longer discussions at Math…”