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Search Results: 1 - 10 of 132200 matches for " Leonid V. Kovalev "
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Symmetric products of the line: embeddings and retractions
Leonid V. Kovalev
Mathematics , 2012,
Abstract: The n-th symmetric product of a metric space is the set of its nonempty subsets with cardinality at most n, equipped with the Hausdorff metric. We prove that every symmetric product of the line is an absolute Lipschitz retract and admits a bi-Lipschitz embedding into a Euclidean space of sufficiently high dimension.
Bi-Lipschitz embedding of projective metrics
Leonid V. Kovalev
Mathematics , 2013, DOI: 10.1090/S1088-4173-2014-00266-5
Abstract: We give a sufficient condition for a projective metric on a subset of a Euclidean space to admit a bi-Lipschitz embedding into Euclidean space of the same dimension.
Sharp distortion growth for bilipschitz extension of planar maps
Leonid V. Kovalev
Mathematics , 2012, DOI: 10.1090/S1088-4173-2012-00243-3
Abstract: This note addresses the quantitative aspect of the bilipschitz extension problem. The main result states that any bilipschitz embedding of $\mathbb R$ into $\mathbb R^2$ can be extended to a bilipschitz self-map of $\mathbb R^2$ with a linear bound on the distortion.
Lipschitz retraction of finite subsets of Hilbert spaces
Leonid V. Kovalev
Mathematics , 2014, DOI: 10.1017/S0004972715000672
Abstract: Finite subset spaces of a metric space $X$ form a nested sequence under natural isometric embeddings $X=X(1)\subset X(2)\subset\dots$. We prove that this sequence admits Lipschitz retractions $X(n)\to X(n-1)$ when $X$ is a Hilbert space.
Variation of quasiconformal mappings on lines
Leonid V. Kovalev,Jani Onninen
Mathematics , 2009, DOI: 10.4064/sm195-3-5
Abstract: We obtain improved regularity of homeomorphic solutions of the reduced Beltrami equation, as compared to the standard Beltrami equation. Such an improvement is not possible in terms of Holder or Sobolev regularity; instead, our results concern the generalized variation of restrictions to lines. Specifically, we prove that the restriction to any line segment has finite p-variation for all p>1 but not necessarily for p=1.
On Invertibility of Sobolev Mappings
Leonid V. Kovalev,Jani Onninen
Mathematics , 2008, DOI: 10.1515/CRELLE.2011.038
Abstract: We prove local and global invertibility of Sobolev solutions of certain differential inclusions which prevent the differential matrix from having negative eigenvalues. Our results are new even for quasiregular mappings in two dimensions.
Quasisymmetric graphs and Zygmund functions
Leonid V. Kovalev,Jani Onninen
Mathematics , 2011, DOI: 10.1007/s11854-012-0039-x
Abstract: A quasisymmetric graph is a curve whose projection onto a line is a quasisymmetric map. We show that this class of curves is related to solutions of the reduced Beltrami equation and to a generalization of the Zygmund class $\Lambda_*$. This relation makes it possible to use the tools of harmonic analysis to construct nontrivial examples of quasisymmetric graphs and of quasiconformal maps.
An N-dimensional version of the Beurling-Ahlfors extension
Leonid V. Kovalev,Jani Onninen
Mathematics , 2009, DOI: 10.5186/aasfm.2011.3620
Abstract: We extend monotone quasiconformal mappings from dimension n to n+1 while preserving both monotonicity and quasiconformality. The extension is given explicitly by an integral operator. In the case n=1 it yields a refinement of the Beurling-Ahlfors extension.
Doubly connected minimal surfaces and extremal harmonic mappings
Tadeusz Iwaniec,Leonid V. Kovalev,Jani Onninen
Mathematics , 2009, DOI: 10.1007/s12220-010-9212-6
Abstract: The concept of a conformal deformation has two natural extensions: quasiconformal and harmonic mappings. Both classes do not preserve the conformal type of the domain, however they cannot change it in an arbitrary way. Doubly connected domains are where one first observes nontrivial conformal invariants. Herbert Groetzsch and Johannes C. C. Nitsche addressed this issue for quasiconformal and harmonic mappings, respectively. Combining these concepts we obtain sharp estimates for quasiconformal harmonic mappings between doubly connected domains. We then apply our results to the Cauchy problem for minimal surfaces, also known as the Bjorling problem. Specifically, we obtain a sharp estimate of the modulus of a doubly connected minimal surface that evolves from its inner boundary with a given initial slope.
Diffeomorphic approximation of Sobolev homeomorphisms
Tadeusz Iwaniec,Leonid V. Kovalev,Jani Onninen
Mathematics , 2010, DOI: 10.1007/s00205-011-0404-4
Abstract: Every homeomorphism h : X -> Y between planar open sets that belongs to the Sobolev class W^{1,p}(X,Y), 1
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