Research

49. Symmetrization and extension of planar bi-Lipschitz maps.
Preprint

48. Uniform convergence of Green’s functions (with Sergei Kalmykov).
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47. On the existence of harmonic mappings between doubly connected domains (with Liulan Li), Proc. Roy. Soc. Edinburgh Sect. A, to appear.
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46. Lipschitz retractions in Hadamard spaces via gradient flow semigroups (with Miroslav Bačák), Canad. Math. Bull. 59 (2016), no. 4, 673–681.
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45. Conformal contractions and lower bounds on the density of harmonic measure, Potential Analysis 46 (2017), no. 2, 385-391.
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44. Lipschitz retraction of finite subsets of Hilbert spaces, Bull. Aust. Math. Soc. 93 (2016), no. 1, 146-151.
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43. Bi-Lipschitz embedding of projective metrics, Conform. Geom. Dyn. 18 (2014), 110-118.
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42. Symmetric products of the line: embeddings and retractions, Proc. Amer. Math. Soc. 143 (2015), 801-809.
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41. Sharp distortion growth for bilipschitz extension of planar maps, Conform. Geom. Dyn. 16 (2012), 124-131.
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Note: The proof of inequality (3.3) contains a slight mistake, but the inequality itself is true. In order to correct the proof, one should replace “Let ρ = dist(ζ, Γj)” at the bottom of page 126 with “Let ρ = dist(ζ, Γ1) – |ζ|”. Then the statement “∂Ω ∩ B(0, ρ) is disjoint from Γ1” is true, and the rest proceeds as written, including the inequality (ρ − |ζ|)/(ρ + |ζ|) ≤ sin(3π/8). Solve this inequality for ρ to get ρ ≤ 29|ζ|, hence dist(ζ, Γ1) ≤ 30|ζ| as is claimed in (3.3).

40. Lipschitz regularity for inner-variational equations (with Tadeusz Iwaniec and Jani Onninen), Duke Math. J. 162 (2013), no. 4, 643-672.
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39. Approximation up to the boundary of homeomorphisms of finite Dirichlet energy (with Tadeusz Iwaniec and Jani Onninen), Bull. London Math. Soc. 44 (2012), no. 5, 871-881.
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38. Quasisymmetric graphs and Zygmund functions (with Jani Onninen), J. Anal. Math. 118 (2012), no. 1, 343-361.
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37. The Hopf-Laplace equation: harmonicity and regularity (with Jan Cristina, Tadeusz Iwaniec, and Jani Onninen), Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13 (2014), no. 4, 1145-1187.
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36. Harmonic mapping problem in the plane (with Jani Onninen), J. Anal. 18 (2010), 279–295.
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35. Diffeomorphic approximation of Sobolev homeomorphisms (with Tadeusz Iwaniec and Jani Onninen), Arch. Rat. Mech. Anal. 201 (2011), no. 3, 1047–1067.
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34. Existence of energy-minimal diffeomorphisms between doubly connected domains (with Tadeusz Iwaniec, Ngin-Tee Koh, and Jani Onninen), Invent. Math. 186 (2011), no. 3, 667–707.
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33. Hopf differentials and smoothing Sobolev homeomorphisms (with Tadeusz Iwaniec and Jani Onninen), Int. Math. Res. Not. IMRN 2012 (2012), no. 14, 3256-3277.
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32. The harmonic mapping problem and affine capacity (with Tadeusz Iwaniec and Jani Onninen), Proc. Roy. Soc. Edinburgh Sect. A 141 (2011), no. 5, 1017–1030.
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Note: A part of the proof of Theorem 1.5, namely Case 1 in section 4.3, misses the possibility that the domain is the complement of the union of a line segment with two half-lines, all lying on the same line. Such a domain (“double Teichmüller ring”) can be treated similarly to the ordinary Teichmüller ring; no new ideas are required. The missing details were later supplied in paper #47 listed above.

31. Doubly connected minimal surfaces and extremal harmonic mappings (with Tadeusz Iwaniec and Jani Onninen), J. Geom. Anal. 22 (2012), no. 3, 726–762.
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30. Area contraction for harmonic automorphisms of the disk (with Ngin-Tee Koh), Bull. London Math. Soc. 43 (2011), no. 1, 91–96.
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29. The Nitsche conjecture (with Tadeusz Iwaniec and Jani Onninen), J. Amer. Math. Soc. 24 (2011), no. 2, 345–373.
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28. Projections and idempotents with fixed diagonal and the homotopy problem for unit tight frames (with Julien Giol, David Larson, Nga Nguyen, and James Tener), Operators and Matrices 5 (2011), no. 1, 139–155.
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27. Invertibility of Sobolev mappings under minimal hypotheses (with Jani Onninen and Kai Rajala), Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), no. 2, 517–528.
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26. An N-dimensional version of the Beurling-Ahlfors extension (with Jani Onninen), Ann. Acad. Sci. Fenn. Math. 36 (2011), 321–329.
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25. Harmonic mappings of an annulus, Nitsche conjecture and its generalizations (with Tadeusz Iwaniec and Jani Onninen), Amer. J. Math. 132 (2010), no. 5, 1397–1428.
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24. Variation of quasiconformal mappings on lines (with Jani Onninen), Studia Math. 195 (2009), no. 3, 257–274.
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Note: A result similar to Proposition 2.1 was proved by Robert Kaufman in Sobolev spaces, dimension, and random series, Proc. Amer. Math. Soc. 128 (2000), no. 2, 427–431.

23. On invertibility of Sobolev mappings (with Jani Onninen), J. Reine Angew. Math. 2011 (2011), no. 656, 1–16.
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22. Dynamics of quasiconformal fields (with Tadeusz Iwaniec and Jani Onninen), J. Dynam. Differential Equations 23 (2011), no. 1, 185–212.
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21. On injectivity of quasiregular mappings (with Tadeusz Iwaniec and Jani Onninen), Proc. Amer. Math. Soc. 137 (2009), no. 5, 1783–1791.
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20. A geometric approach to accretivity, Studia Math. 181 (2007), no. 1, 87–100.
Journal

19. Doubling measures, monotonicity, and quasiconformality (with Diego Maldonado and Jang-Mei Wu), Math. Z. 257 (2007), no. 3, 525–545.
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18. Convex functions and quasiconformal mappings (with Diego Maldonado), in Harmonic analysis, partial differential equations, and related topics, 93–104, Contemporary Math., vol. 428, Amer. Math. Soc., 2007.
Proceedings Preprint

17. Hyperbolic and quasisymmetric structure of hyperspaces (with Jeremy Tyson), in In the tradition of Ahlfors-Bers, IV, 151–166, Contemporary Math., vol. 432, Amer. Math. Soc., 2007.
Proceedings Preprint

16. Quasiconformal geometry of monotone mappings, J. London Math. Soc. 75 (2007), no. 2, 391–408.
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15. Conformal dimension does not assume values between zero and one, Duke Math. J. 134 (2006), no. 1, 1–13.
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14. Mappings with convex potentials and the quasiconformal Jacobian problem (with Diego Maldonado), Illinois J. Math. 49 (2005), no. 4, 1039–1060.
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13. On Hölder regularity for elliptic equations of non-divergence type in the plane (with Albert Baernstein), Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), 295–317.
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12. Comparison theorems for the one-dimensional Schrödinger equation, Ark. Mat. 43 (2005), no. 2, 403–418.
Journal

11. Quasiregular gradient mappings and strong solutions of elliptic equations (with David Opěla), in The p-harmonic equation and recent advances in analysis, 145–157, Contemporary Math. vol. 370, AMS, 2005.
Proceedings Preprint

10. On G-compactness of the Beltrami operators (with Flavia Giannetti, Tadeusz Iwaniec, Gioconda Moscariello, and Carlo Sbordone), in Nonlinear homogenization and its applications to composites, polycrystals and smart materials, 107–138, NATO Science Series II, vol. 170, Kluwer, 2004.

9. Hölder spaces of quasiconformal mappings, Publ. Inst. Math. (Beograd) 75 (89) (2004), 87–94.
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8. Quasiregular mappings of maximal local modulus of continuity, Ann. Acad. Sci. Fenn. Math. 29 (2004), 211–222.
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7. Boundary values of mappings of finite distortion (with Jani Onninen), Rep. Univ. Jyväskylä Dep. Math. Stat. 92 (2003), 175–182.
Preprint

6. Monotonicity of the generalized reduced modulus, J. Math. Sci., New York 118 (2003), no.1, 4861–4870; translation from Zap. Nauchn. Sem. S-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 276 (2001), Anal. Teor. Chisel i Teor. Funkts. 17, 219–236.
In Russian In English

5. Estimates of conformal radius and distortion theorems for univalent functions, J. Math. Sci., New York 110 (2002), no. 6, 3111–3120; translation from Zap. Nauchn. Sem. S-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 263 (2000), Anal. Teor. Chisel i Teor. Funkts. 16, 141–156.
In Russian In English

4. Domains with convex hyperbolic radius, Acta Math. Univ. Comenianae 70 (2001), no. 2, 207–213.
Journal

3. The reduced modulus of the complex sphere (with Vladimir Dubinin), J. Math. Sci., New York 105 (2001), no. 4, 2165–2179; translation from Zap. Nauchn. Sem. S-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 254 (1998), Anal. Teor. Chisel i Teor. Funkts. 15, 76–94.
In Russian In English

2. On the inner radii of symmetric nonoverlapping domains, Russian Math. (Iz. VUZ) 44 (2000), no. 6, 77–78; translation from Izv. Vyssh. Uchebn. Zaved. Mat. (2000), no. 6, 80–81.
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1. On the problem of extremal partitioning with free poles on the circle, Dal’nevost. Mat. Sb. 2 (1996), 96–98. (In Russian).
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