Dimension gap under Sobolev mappings

We prove an essentially sharp estimate in terms of generalized Hausdorff measures for the images of boundaries of Hölder domains under continuous Sobolev mappings, satisfying suitable Orlicz–Sobolev conditions. This estimate marks a dimension gap, which was first observed in [2] for conformal mappings.     

A topological aspect of Sobolev mappings

Let be a connected open set, . We give a sufficient condition for a mapping , , to have the property that sgn is almost everywhere of one sign. Following the work of Müller, Spector, and Tang [MST], we give an application of our results to the theory of non-linear elasticity. Received: 13 October 2000 / Accepted: 23 January 2001 / Published online: 4 May 2001     

On global singularities of Sobolev mappings

We define various invariants for Sobolev mappings defined between manifolds which are stable under perturbation with respect to the strong Sobolev topology. We show that these invariants classify various types of ``global singularities" of Sobolev maps. These invariants are used to give a simple characterization of the strong closure of the set of smooth maps in the Sobolev space.     

Extensions for Sobolev mappings between manifolds

We consider two compact Riemannian manifoldsM andN, such thatM has a boundary (but notN). We address the extension problem in the Sobolev class, namely, we investigate the question: foru W ^1–1/p,pM,N is there a mapV inW ^1/p(M,N) such thatV=u on M. Various results are given, and an emphasis is put on the special (simple) caseN=S ¹.     

The co-area formula for Sobolev mappings

We extend Federer's co-area formula to mappings belonging to the Sobolev class , , m$">, and more generally, to mappings with gradient in the Lorentz space . This is accomplished by showing that the graph of in is a Hausdorff -rectifiable set.

     

Estimates for Sobolev norms of triangular mappings

An increasing triangular mapping T on the n-dimensional cube Θ = [0, 1] ⁿ transforming a measure μ to a measure ν is considered, where μ and ν are absolutely continuous Borel probability measures having densities ρ _μ and ρ _ν . It is shown that if there exist positive constants ? and M such that ? < ρ _ν < M, ? < ρ _ν < M, there exist numbers α, β > 1 such that p = αβ(n ? 1)^?1 (α + β)^?1 > 1 and ρ _μ ∈ W ^1,α (Θ), ρ _ν ∈ W ^1,β ) (Θ), where W ^1,q denotes a Sobolev class, then the mapping T belongs to the class W ^1,p (Θ).     

Invertibility of Sobolev mappings under minimal hypotheses

We prove a version of the Inverse Function Theorem for continuous weakly differentiable mappings. Namely, a nonconstant W1,n$W^{1,n}$ mapping is a local homeomorphism if it has integrable inner distortion function and satisfies a certain differential inclusion. The integrability assumption is shown to be optimal.     

Density of Lipschitz mappings in the class of Sobolev mappings between metric spaces

We prove that Lipschitz mappings are dense in the Newtonian–Sobolev classes N ^1,p (X, Y) of mappings from spaces X supporting p-Poincaré inequalities into a finite Lipschitz polyhedron Y if and only if Y is [p]-connected, π ₁(Y) = π ₂(Y) = · · · = π _[p](Y) = 0, where [p] is the largest integer less than or equal to p. This work was supported by the NSF grant DMS-0500966.     

Sobolev classes of Banach space-valued functions and quasiconformal mappings

We give a definition for the class of Sobolev functions from a metric measure space into a Banach space. We give various characterizations of Sobolev classes and study the absolute continuity in measure of Sobolev mappings in the “borderline case”. We show under rather weak assumptions on the source space that quasisymmetric homeomorphisms belong to a Sobolev space of borderline degree; in particular, they are absolutely continuous. This leads to an analytic characterization of quasiconformal mappings between Ahlfors regular Loewner spaces akin to the classical Euclidean situation. As a consequence, we deduce that quasisymmetric maps respect the Cheeger differentials of Lipschitz functions on metric measure spaces with borderline Poincaré inequality. J. H. supported by NSF grant DMS9970427. P. K. supported by the Academy of Finland, project 39788. N. S. supported in part by Enterprise Ireland. J. T. T. supported by an NSF Postdoctoral Research Fellowship.     

Isomorphisms of Sobolev spaces on Carnot groups and quasi-isometric mappings

We study the properties of the mappings on a Carnot group which induce, via the change-of-variables rule, the isomorphisms of Sobolev spaces with the summability exponent different from the Hausdorff dimension of the group.     

Hölder continuity of Sobolev functions and quasiconformal mappings

Part of the research for this paper was done while the first author was visiting at the Mittag-Leffler Institute. He wishes to express his gratitude to the Institute for its hospitality     

Generalized Hausdorff dimension distortion in Euclidean spaces under Sobolev mappings

We investigate how the integrability of the derivatives of Orlicz-Sobolev mappings defined on open subsets of Rn affect the sizes of the images of sets of Hausdorff dimension less than n. We measure the sizes of the image sets in terms of generalized Hausdorff measures.     

Sard's theorem for mappings in Hölder and Sobolev spaces

We prove various generalizations of classical Sard's theorem to mappings f:M ^m →N ⁿ between manifolds in Hölder and Sobolev classes. It turns out that if f ∈ C ^k,λ (M ^m ,N ⁿ ), then—for arbitrary k and λ—one can obtain estimates of the Hausdorff measure of the set of critical points in a typical level set f ^?1(y). The classical theorem of Sard holds true for f ∈ C ^k with sufficiently large k, i.e., k>max(m?n,0); our estimates contain Sard's theorem (and improvements due to Dubovitskii and Bates) as special cases. For Sobolev mappings between manifolds, we describe the structure of f ^?1(y).     

Randomized approximation of Sobolev embeddings, III

We continue the study of randomized approximation of embeddings between Sobolev spaces on the basis of function values. The source space is a Sobolev space with nonnegative smoothness order; the target space has negative smoothness order. The optimal order of approximation (in some cases only up to logarithmic factors) is determined. Extensions to Besov and Bessel potential spaces are given and a problem recently posed by Novak and Woźniakowski is partially solved. The results are applied to the complexity analysis of weak solution of elliptic PDE.