0000000001044949
AUTHOR
Ville Tengvall
Mappings of Finite Distortion : Compactness of the Branch Set
We show that an entire branched cover of finite distortion cannot have a compact branch set if its distortion satisfies a certain asymptotic growth condition. We furthermore show that this bound is strict by constructing an entire, continuous, open and discrete mapping of finite distortion which is piecewise smooth, has a branch set homeomorphic to an (n - 2)-dimensional torus and distortion arbitrarily close to the asymptotic bound. Peer reviewed
Mappings of Lp-integrable distortion: regularity of the inverse
Let be an open set in ℝn and suppose that is a Sobolev homeomorphism. We study the regularity of f–1 under the Lp-integrability assumption on the distortion function Kf. First, if is the unit ball and p > n – 1, then the optimal local modulus of continuity of f–1 is attained by a radially symmetric mapping. We show that this is not the case when p ⩽ n – 1 and n ⩾ 3, and answer a question raised by S. Hencl and P. Koskela. Second, we obtain the optimal integrability results for ∣Df–1∣ in terms of the Lp-integrability assumptions of Kf.
Approximation of W1,p Sobolev homeomorphism by diffeomorphisms and the signs of the Jacobian
Let Ω ⊂ R n, n ≥ 4, be a domain and 1 ≤ p 0 on a set of positive measure and Jf < 0 on a set of positive measure. It follows that there are no diffeomorphisms (or piecewise affine homeomorphisms) fk such that fk → f in W1,p . peerReviewed
On proper branched coverings and a question of Vuorinen
We study global injectivity of proper branched coverings from the open Euclidean n$n$-ball onto an open subset of the Euclidean n$n$-space in the case where the branch set is compact. In particular, we show that such mappings are homeomorphisms when n=3$n=3$ or when the branch set is empty. This gives a positive answer to the corresponding cases of a question of Vuorinen. Peer reviewed
Differentiability in the Sobolev space W1,n-1
Let Ω ⊂ Rn be a domain, n ≥ 2. We show that a continuous, open and discrete mapping f ∈ W1,n−1 loc (Ω, Rn ) with integrable inner distortion is differentiable almost everywhere on Ω. As a corollary we get that the branch set of such a mapping has measure zero. peerReviewed
On BLD-mappings with small distortion
We show that every $$L$$ -BLD-mapping in a domain of $$\mathbb {R}^{n}$$ is a local homeomorphism if $$L < \sqrt{2}$$ or $$K_I(f) < 2$$ . These bounds are sharp as shown by a winding map.
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.
Remarks on Martio’s conjecture
We introduce a certain integrability condition for the reciprocal of the Jacobian determinant whichguarantees the local homeomorphism property of quasiregular mappings with a small inner dilata-tion. This condition turns out to be sharp in the planar case. We also show that every branch pointof a quasiregular mapping with a small inner dilatation is a Lebesgue point of the differentialmatrix of the mapping. peerReviewed
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 L p -integrable distortion: regularity of the inverse
Let X be an open set in R n and suppose that f : X → R n is a Sobolev homeomorphism. We study the regularity of f −1 under the L p -integrability assumption on the distortion function Kf . First, if X is the unit ball and p > n−1, then the optimal local modulus of continuity of f −1 is attained by a radially symmetric mapping. We show that this is not the case when p 6 n − 1 and n > 3, and answer a question raised by S. Hencl and P. Koskela. Second, we obtain the optimal integrability results for |Df −1 | in terms of the L p -integrability assumptions of Kf . peerReviewed
Absolute continuity of mappings with finite geometric distortion
Suppose that ⊂ R n is a domain with n ≥ 2. We show that a continuous, sense-preserving, open and discrete mapping of finite geometric outer distortion with KO(·,f) ∈ L 1/(n 1) loc () is absolutely continuous on almost every line parallel to the coordinate axes. Moreover, if U ⊂ is an open set with N(f,U) 0 depends only on n and on the maximum multiplicity N(f,U).
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