We give an intrinsic characterization of the restrictions of Sobolevspaces to regular subsets of R n via sharp maximal functions and local approximations.

No abstract

No abstract

Let F be a set-valued mapping which to each point x of a metric space (M, ρ) assigns a convex closed set F(x) ⊂ R 2 . We present several constructive criteria for the existence of a Lipschitz selection of F, i.e., a Lipschitz mapping f : M → R 2 such that f (x) ∈ F(x) for every x ∈ M. The geometric methods we develop to prove these criteria provide efficient algorithms for constructing nearly optimal Lipschitz selections and computing the order of magnitude of their Lipschitz seminorms. Contents 1. Introduction. 2. Notation and preliminaries. 2.1 Background notation. 2.2 Rectangles and rectangular hulls. 2.3 Rectangles: intersections, neighborhoods and selections. 3. The key theorem: Lipschitz selections and rectangular hulls. 4. Proof of the key theorem: the final step. 5. Lipschitz selection criteria in the two dimensional case. 5.1 Constructive criteria for Lipschitz selections: proofs. 5.2 Criteria for Lipschitz selections in terms of intersections of sets. 6. Projection Algorithm for nearly optimal Lipschitz selections. 6.1 The ⃗ λ-Projection Algorithm. 6.2 Projection Algorithms and a solution to the second main problem. 6.3 The constant Λ R (F) and other related constants. 6.4 Lipschitz selections of polygon-set valued mappings. 7. Lipschitz selections and iterations of balanced refinements. 7.1 The Stabilization Principle for balanced refinements of set-valued mappings. 7.2 The Iterative Algorithm for set-valued mappings. References

Abstract. We prove that the trace of the space C 1,ω (R n ) to an arbitrary closed subset X ⊂ R n is characterized by the following "finiteness" property. A function f : X → R belongs to the trace space if and only if the restriction f | Y to an arbitrary subset Y ⊂ X consisting of at most 3·2 n−1 can be extendedThe constant 3 · 2 n−1 is sharp. The proof is based on a Lipschitz selection result which is interesting in its own right.