An Aronsson type approach to extremal quasiconformal mappings

Journal of Differential Equations - Tập 253 Số 3 - Trang 851-877 - 2012
Luca Capogna1,2, Andrew Raich1
1Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR 72701, United States
2Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455, United States

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Adamowicz, 2007, The Grötzsch problem in higher dimensions, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 18, 163, 10.4171/RLM/488

Ahlfors, 1954, On quasiconformal mappings, J. Anal. Math., 3, 1, 10.1007/BF02803585

Ahlfors, 1974, Conditions for quasiconformal deformations in several variables, 19

Ahlfors, 1976, Quasiconformal deformations and mappings in Rn, J. Anal. Math., 30, 74, 10.1007/BF02786705

Ambrosio, 2004, Transport equation and Cauchy problem for BV vector fields, Invent. Math., 158, 227, 10.1007/s00222-004-0367-2

Aronsson, 1967, Extension of functions satisfying Lipschitz conditions, Ark. Mat., 6, 551, 10.1007/BF02591928

Aronsson, 1968, On the partial differential equation ux2uxx+2uxuyuxy+uy2uyy=0, Ark. Mat., 7, 395, 10.1007/BF02590989

Asadchiĭ, 1991, On the maximum principle for n-dimensional quasiconformal mappings, Mat. Zametki, 50, 14

Astala, 2010, Deformations of annuli with smallest mean distortion, Arch. Ration. Mech. Anal., 195, 899, 10.1007/s00205-009-0231-z

Astala, 2005, Extremal mappings of finite distortion, Proc. Lond. Math. Soc. (3), 91, 655, 10.1112/S0024611505015376

Balogh, 2011, Modulus of curve families and extremality of spiral-stretch maps, J. Anal. Math., 113, 265, 10.1007/s11854-011-0007-x

Barron, 2001, Lower semicontinuity of L∞ functionals, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18, 495, 10.1016/S0294-1449(01)00070-1

Bers, 1960, Quasiconformal mappings and Teichmüllerʼs theorem, 89

Bonk, 2008, Logarithmic potentials, quasiconformal flows, and Q-curvature, Duke Math. J., 142, 197, 10.1215/00127094-2008-005

Crandall, 1992, Userʼs guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27, 1, 10.1090/S0273-0979-1992-00266-5

Dacorogna, 2006, Extension theorems for vector valued maps, J. Math. Pures Appl. (9), 85, 313, 10.1016/j.matpur.2005.04.005

DiPerna, 1989, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98, 511, 10.1007/BF01393835

Evans, 2005, Diffeomorphisms and nonlinear heat flows, SIAM J. Math. Anal., 37, 737, 10.1137/04061386X

Fehlmann, 1987, Extremal problems for quasiconformal mappings in space, J. Anal. Math., 48, 179, 10.1007/BF02790328

Gehring, 2005, Quasiconformal mappings in Euclidean spaces, 1, 10.1016/S1874-5709(05)80005-8

Gehring, 1965, The coefficients of quasiconformality of domains in space, Acta Math., 114, 1, 10.1007/BF02391817

Giaquinta, 1983, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, vol. 105

Grötzsch, 1928, Über die Verzerrung bei schlichten nichtkonformen Abbildungen und über eine damit zusammenhängende Erweiterung des Picardschen Satzes, Ber. Leipzig, 80, 503

Hamilton, 1969, Extremal quasiconformal mappings with prescribed boundary values, Trans. Amer. Math. Soc., 138, 399, 10.1090/S0002-9947-1969-0245787-3

Iwaniec, 2001, Geometric Function Theory and Non-Linear Analysis, 10.1093/oso/9780198509295.001.0001

Jensen, 1988, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Ration. Mech. Anal., 101, 1, 10.1007/BF00281780

H. Koch, T. Lamm, Geometric flows with rough initial data, preprint, 2009.

Misawa, 2004, Existence of a classical solution for linear parabolic systems of nondivergence form, Comment. Math. Univ. Carolin., 45, 475

Müller, 2005, Parabolic systems with nowhere smooth solutions, Arch. Ration. Mech. Anal., 177, 1, 10.1007/s00205-004-0352-3

Reimann, 1976, Ordinary differential equations and quasiconformal mappings, Invent. Math., 33, 247, 10.1007/BF01404205

Sarvas, 1979, Ahlforsʼ trivial deformations and Liouvilleʼs theorem in Rn, vol. 747, 343

Schlag, 1996, Schauder and Lp estimates for parabolic systems via Campanato spaces, Comm. Partial Differential Equations, 21, 1141, 10.1080/03605309608821221

Semenov, 1976, One-parameter groups of quasiconformal homeomorphisms in a Euclidean space, Sibirsk. Mat. Zh., 17, 177

Semenov, 1980, Necessary conditions in extremal problems for spatial quasiconformal mappings, Sibirsk. Mat. Zh., 21

Semenov, 1981, On sufficient conditions for extremal quasiconformal mappings in space, Sibirsk. Mat. Zh., 22

Semënov, 1990, Some extremal problems in the theory of quasiconformal mappings, Sibirsk. Mat. Zh., 31

S. Sheffield, C.K. Smart, Vector-valued optimal Lipschitz extensions, preprint, 2010.

Strebel, 1986, Extremal quasiconformal mappings, Results Math., 10, 168, 10.1007/BF03322374

Teichmüller, 1940, Extremale quasikonforme Abbildungen und quadratische Differentiale, Abh. Preuss. Akad. Wiss. Math.-Nat. Kl., 1939, 197

Väisälä, 1971, Lectures on n-Dimensional Quasiconformal Mappings, vol. 229