A Note on Extremal Mappings of Finite Distortion

Sharp generalized Trudinger inequalities via truncation

Abstract We prove that the generalized Trudinger inequalities into exponential and double exponential Orlicz spaces improve to inequalities on Orlicz–Lorentz spaces provided they are stable under truncation.

Explicit Upper Bound for Entropy Numbers

We give an explicit upper bound for the entropy numbers of the embedding I : W r,p(Ql) → C(Ql) where Ql = (−l, l)m ⊂ Rm, r ∈ N, p ∈ (1,∞) and rp > m.

Composition of quasiconformal mappings and functions in Triebel-Lizorkin spaces

Let α > 0 and p ∈ [1, ∞) satisfy αp ≤ n. Suppose that f: Rn Rn is a K-quasiconformal mapping and let u ∈ Wα, p(Rn) have compact support. We find an optimal value of β = β(α, K, n) such that u○f ∈ Wβ, p(Rn). We also give an answer to the analogous problem where we moreover assume that u is bounded.

Images and Preimages of Null Sets

In this chapter we study conditions that guarantee that our mapping maps sets of measure zero to sets of measure zero. We start with the problem in general Sobolev spaces, after which we establish a better result for mappings of finite distortion. Then we introduce a natural class of counterexamples to statements of this type and finally we give a weak condition under which the preimage of a set of measure zero has measure zero for mappings of finite distortion.

Openness and Discreteness

The aim of this chapter is to study conditions under which a mapping of finite distortion is open (maps open sets to open sets) and discrete (preimage of each point is a discrete set).

Homeomorphisms of Finite Distortion

In this chapter we establish the optimal regularity of the inverse mapping in higher dimensions and optimal Sobolev regularity for composites. Moreover, we establish optimal moduli of continuity for mappings in our classes and we discuss orientation preservation and approximation of Sobolev homeomorphisms.

Mappings of finite distortion : size of the branch set

Abstract We study the branch set of a mapping between subsets of ℝ n {\mathbb{R}^{n}} , i.e., the set where a given mapping is not defining a local homeomorphism. We construct several sharp examples showing that the branch set or its image can have positive measure.

Regularity of the inverse of a Sobolev homeomorphism in space

Let Ω ⊂ Rn be open. Given a homeomorphism of finite distortion with |Df| in the Lorentz space Ln−1, 1 (Ω), we show that and f−1 has finite distortion. A class of counterexamples demonstrating sharpness of the results is constructed.

Mappings of finite distortion: discreteness and openness for quasi-light mappings

Abstract Let f ∈ W 1 , n ( Ω , R n ) be a continuous mapping so that the components of the preimage of each y ∈ R n are compact. We show that f is open and discrete if | D f ( x ) | n ⩽ K ( x ) J f ( x ) a.e. where K ( x ) ⩾ 1 and K n − 1 / Φ ( log ( e + K ) ) ∈ L 1 ( Ω ) for a function Φ that satisfies ∫ 1 ∞ 1 / Φ ( t ) d t = ∞ and some technical conditions. This divergence condition on Φ is shown to be sharp.

Approximation of W1, Sobolev homeomorphism by diffeomorphisms and the signs of the Jacobian

Abstract Let Ω ⊂ R n , n ≥ 4 , be a domain and 1 ≤ p [ n / 2 ] , where [ a ] stands for the integer part of a. We construct a homeomorphism f ∈ W 1 , p ( ( − 1 , 1 ) n , R n ) such that J f = det ⁡ D f > 0 on a set of positive measure and J f 0 on a set of positive measure. It follows that there are no diffeomorphisms (or piecewise affine homeomorphisms) f k such that f k → f in W 1 , p .

Mappings of finite distortion: Reverse inequalities for the Jacobian

Let f be a nonconstant mapping of finite distortion. We establish integrability results on 1/Jf by studying weights that satisfy a weak reverse Holder inequality where the associated constant can depend on the ball in question. Here Jf is the Jacobian determinant of f.

Quasihyperbolic boundary conditions and capacity: Uniform continuity of quasiconformal mappings

We prove that quasiconformal maps onto domains which satisfy a suitable growth condition on the quasihyperbolic metric are uniformly continuous when the source domain is equipped with the internal metric. The obtained modulus of continuity and the growth assumption on the quasihyperbolic metric are shown to be essentially sharp. As a tool, we prove a new capacity estimate.

Integrability of J f and 1∕J f

In this chapter we study the optimal degree of integrability of J f and 1∕J f for mappings of finite distortion. As an application of our estimates we show that some sets are removable singularities for mappings with exponentially integrable distortion.

Dimension gap under conformal mappings

Abstract We give an estimate for the Hausdorff gauge dimension of the boundary of a simply connected planar domain under p -integrability of the hyperbolic metric, p > 1 . This estimate does not degenerate when p tends to one; for p = 1 the boundary can even have positive area. The same phenomenon is extended to general planar domains in terms of the quasihyperbolic metric. We also give an example which shows that our estimates are essentially sharp.

Jacobian of weak limits of Sobolev homeomorphisms

Abstract Let Ω be a domain in ℝ n {\mathbb{R}^{n}} , where n = 2 , 3 {n=2,3} . Suppose that a sequence of Sobolev homeomorphisms f k : Ω → ℝ n {f_{k}\colon\Omega\to\mathbb{R}^{n}} with positive Jacobian determinants, J ⁢ ( x , f k ) > 0 {J(x,f_{k})>0} , converges weakly in W 1 , p ⁢ ( Ω , ℝ n ) {W^{1,p}(\Omega,\mathbb{R}^{n})} , for some p ⩾ 1 {p\geqslant 1} , to a mapping f. We show that J ⁢ ( x , f ) ⩾ 0 {J(x,f)\geqslant 0} a.e. in Ω. Generalizations to higher dimensions are also given.

Sharpness of the differentiability almost everywhere and capacitary estimates for Sobolev mappings

We give sharp conformal conditions for the dfferentiability in the Sobolev space W1, n-1 loc (Ω,Rn). Furthermore, we show that the space W1, n-1 loc (Ω,Rn) can be considered as the borderline space for some capacitary inequalities. peerReviewed