On strong solutions of the Beltrami equations

References

• Andreian Cazacu , C . 1971 . Influence of the Orientation of the Characteristic Ellipses on the Properties of the Quasiconformal Mappings . Proceedings of Romanian–Finnish Seminar on Teichmüller Spaces and Riemann Surfaces . 1971 , Bucharest . pp. 65 – 85 .
   Google Scholar
• Andreian Cazacu , C . 1976 . Moduli inequalities for quasiregular mappings . Ann. Acad. Sci. Fenn. Math. , 2 : 17 – 28 .
   Google Scholar
• Ryazanov , V , Srebro , U and Yakubov , E . 2005 . On ring solutions of Beltrami equations . J. d'Analyse Math. , 96 : 117 – 150 .
   Web of Science ®Google Scholar
• Gutlyanskii , V , Martio , O , Sugawa , T and Vuorinen , M . 2005 . On the degenerate Beltrami equation . Trans. Amer. Math. Soc. , 357 : 875 – 900 .
   Web of Science ®Google Scholar
• Andreian Cazacu , C . 1957 . Sur les transformations pseudo-analytiques . Revue Math. Pures Appl. , 2 : 383 – 397 .
   Google Scholar
• Andreian Cazacu , C . 1956 . “ Sur les ralations entre les functions caracteristiques de la pseudo–analyticite ” . In in Lucrarile celui de al IV-lea Congres al Matematicienilor , Bucuresti : Romani .
   Google Scholar
• Andreian Cazacu , C . 2005 . On the length-area dilatation . Complex Var. , 50 ( 7–11 ) : 765 – 776 .
   Google Scholar
• Ahlfors , LV . 1966 . Lectures on Quasiconformal Mappings , Princeton : D. Van Nostrand Company .
   Google Scholar
• Belinskii , PP . 1974 . General properties of quasiconformal mappings , Novosibirsk : Izdat. ‘Nauka’ Sibirsk. Otdel. . (Russian)
   Google Scholar
• Bojarski , B . 1957 . Generalized solutions of a system of differential equations of the first order of the elliptic type with discontinuous coefficients . Mat. Sb. , 43 ( 4,85 ) : 451 – 503 . (Russian)
   Google Scholar
• Lehto , O and Virtanen , K . 1973 . Quasiconformal Mappings in the Plane , New York : Springer .
   Google Scholar
• Astala , K , Iwaniec , T and Martin , G . 2009 . Elliptic differential equations and quasiconformal mappings in the plane , Princeton : Pronceton Univ. Press .
   Google Scholar
• Bojarski , B , Gutlyanskii , V and Ryazanov , V . 2008 . General Beltrami equations and BMO . Ukrainian Math. Bull. , 5 ( 3 ) : 305 – 326 .
   Google Scholar
• Bojarski , B , Gutlyanskii , V and Ryazanov , V . 2009 . On the Beltramin equations with two characteristics . Complex Variables and Elliptic Equations , 54 ( 10 ) : 935 – 950 .
   Web of Science ®Google Scholar
• Brakalova , MA and Jenkins , JA . 1998 . On solutions of the Beltrami equation . J. Anal. Math. , 76 : 67 – 92 .
   Web of Science ®Google Scholar
• Brakalova , MA and Jenkins , JA . 2004 . On solutions of the Beltrami equation: II . Publ. de l'Inst. Math. , 75 ( 89 ) : 3 – 8 .
   Google Scholar
• Chen , ZG . 2003 . μ(z)-homeomorphisms of the plane . Michigan Math. J. , 51 ( 3 ) : 547 – 556 .
   Web of Science ®Google Scholar
• David , G . 1988 . Solutions de l'equation de Beltrami avec ∥μ∥∞ = 1 . Ann. Acad. Sci. Fenn. Ser. AI. Math. AI. , 13 ( 1 ) : 25 – 70 .
   Google Scholar
• Iwaniec , T and Martin , G . 2008 . The Beltrami equation . Memories of AMS , 191 : 1 – 92 .
   Web of Science ®Google Scholar
• Kruglikov , VI . 1973 . “ The existence and uniqueness of mappings that are quasiconformal in the mean ” . In in Metric Questions of the Theory of Functions and Mappings , 123 – 147 . Kiev : Naukova Dumka . (Russain)
   Google Scholar
• Lehto , O . 1968 . Homeomorphisms with a prescribed dilatation . Lecture Notes in Math. , 118 : 58 – 73 .
   Google Scholar
• Martio , O and Miklyukov , V . 2004 . On existence and uniqueness of the degenerate Beltrami equation . Complex Variables Theory Appl. , 49 : 647 – 656 .
   Google Scholar
• Martio , O , Miklyukov , V and Vuorinen , M . 2005 . Some remarks on an existence problem for degenerate elliptic system . Proc. Amer. Math. Soc. , 133 : 1451 – 1458 .
   Web of Science ®Google Scholar
• Martio , O , Ryazanov , V , Srebro , U and Yalubov , E . 2009 . Moduli in modern mapping theory , New York : Springer .
   Google Scholar
• Miklyukov , VM and Suvorov , GD . 1972 . “ On existence and uniqueness of quasiconformal mappings with unbounded characteristics ” . In in Investigations in the Theory of Functions of Complex Variables and its Applications , Kiev : Institute of Mathematics .
   Google Scholar
• Pesin , IN . 1969 . Mappings quasiconformal in the mean . Dokl. Akad. Nauk SSSR , 187 ( 4 ) : 740 – 742 .
   Google Scholar
• Tukia , P . 1991 . Compactness properties of μ-homeomorphisms . Ann. Acad. Sci. Fenn. Ser. AI. Math. AI. , 16 ( 1 ) : 47 – 69 .
   Google Scholar
• Yakubov , E . 1978 . Solutions of Beltrami's equation with degeneration . Dokl. Akad. Nauk SSSR , 243 ( 5 ) : 1148 – 1149 .
   Google Scholar
• Srebro , U and Yakubov , E . 2005 . The Beltrami Equation, Handbook in Complex Analysis: Geometric Function Theory , Vol. 2 , 555 – 597 . Amsterdam : Elseiver B.V. .
   Google Scholar
• Ryazanov , V , Srebro , U and Yakubov , E . 2001 . BMO-quasiconformal mappings . J. d’Analyse Math. , 83 : 1 – 20 .
   Web of Science ®Google Scholar
• Ryazanov , V , Srebro , U and Yakubov , E . 2006 . Finite mean oscillation and the Beltrami equation . Israel J. Math. , 153 : 247 – 266 .
   Web of Science ®Google Scholar
• Ignat'ev , A and Ryazanov , V . 2005 . Finite mean oscillation in the mapping theory . Ukrainian Math. Bull. , 2 ( 3 ) : 403 – 424 .
   Google Scholar
• Ryazanov , V , Srebro , U and Yakubov , E . 2003 . Degenerate Beltrami equation and radial Q-homeomorphisms . Reports Math. Dept. Helsinki Univ. , 369 : 1 – 34 .
   Google Scholar
• Reich , E and Walczak , H . 1965 . On the behaviour of quasiconformal mappings at a point . Trans. Amer. Math. Soc. , 117 : 338 – 351 .
   Web of Science ®Google Scholar
• Maz’ya , VG and Poborchi , SV . 1997 . Differentiable Functions on Bad Domains , Singapure, New Jersey, London, Hong Kong : World Scientific .
   Google Scholar
• Gehring , FW . 1962 . Rings and quasiconformal mappings in space . Trans. Amer. Math. Soc. , 103 : 353 – 393 .
   Google Scholar
• Salimov , R . 2008 . ACL and differentiability of Q-homeomorphisms . Ann. Acad. Sci. Fenn. Math. , 33 : 295 – 301 .
   Google Scholar
• Ponomarev , SP . 1995 . The N −1–property of mappings, and Lusin's (N) condition . Mat. Zametki , 58 : 411 – 418 . translation in Math. Notes 58 (1995), pp. 960–965
   Google Scholar
• Hencl , S and Koskela , P . 2006 . Regularity of the inverse of a planar Sobolev homeomorphisms . Arch. Ration Mech. Anal. , 180 ( 1 ) : 75 – 95 .
   Web of Science ®Google Scholar
• Ryazanov , V and Salimov , R . 2007 . Weakly flat spaces and boudaries in the mapping theory . Ukr. Mat. Vis. , 4 ( 2 ) : 199 – 234 . [in Russian] (translation in Ukrain. Math. Bull. 4(2) (2007), pp. 199–233)
   Google Scholar
• Reimann , HM and Rychener , T . 1975 . Funktionen Beschrankter Mittlerer Oscillation , Berlin : Springer .
   Google Scholar
• John , F and Nirenberg , L . 1961 . On functions of bounded mean oscillation . Commun. Pure Appl. Math. , 14 : 415 – 426 .
   Web of Science ®Google Scholar
• Sarason , D . 1975 . Functions of vanishing mean oscillation . Trans. Amer. Math. Soc. , 207 : 391 – 405 .
   Web of Science ®Google Scholar
• Chiarenza , F , Frasca , M and Longo , P . 1993 . W 2,p -solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients . Trans. Amer. Math. Soc. , 336 ( 2 ) : 841 – 853 .
   Web of Science ®Google Scholar
• Iwaniec , T and Sbordone , C . 1998 . Riesz transforms and elliptic PDEs with VMO coefficients . J. Anal. Math. , 74 : 183 – 212 .
   Web of Science ®Google Scholar
• Martio , O , Ryazanov , V and Vuorinen , M . 1999 . BMO and injectivity of space quasiregular mappings . Math. Nachr. , 205 : 149 – 161 .
   Web of Science ®Google Scholar
• Palagachev , DK . 1995 . Quasilinear elliptic equations with VMO coefficients . Trans. Amer. Math. Soc. , 347 ( 7 ) : 2481 – 2493 .
   Web of Science ®Google Scholar
• Ragusa , MA . 1999 . Elliptic boundary value problem in vanishing mean oscillation hypothesis . Comment. Math. Univ. Carolin. , 40 ( 4 ) : 651 – 663 .
   Google Scholar
• Heinonen , J and Koskela , P . 1993 . Sobolev mappings with integrable dilatations . Arch. Rational Mech. Anal. , 125 : 81 – 97 .
   Web of Science ®Google Scholar
• Sobolev , SL . 1950; . Applications of functional analysis in mathematical physics , Leningrad : Izdat. Gos. Univ. . English transl, Amer. Math. Soc., Providence, RI, 1963
   Google Scholar
• Dunford , N and Schwartz , JT . 1957 . Linear Operators, Part I: General Theory , New York, London : Interscience Publishers .
   Google Scholar
• Hörmander , L . 1983 . The Analysis of Linear Partial Differential Operators: I. Distribution Theory and Fourier Analysis , Springer : Berlin .
   Google Scholar
• Reshetnyak , YuG . 1989 . Space Mappings with Bounded Distortion . Transl. of Math. Monographs , 73 Providence, AMS
   Google Scholar
• Evans , LC and Gapiery , RF . 1992 . Measure Theory and Fine Properties of Functions , Boca Raton , FL : CRC Press .
   Google Scholar
• Saks , S . 1964 . Theory of the Integral , New York : Dover Publications .
   Google Scholar
• Dugundji , J . 1966 . Topology , Boston : Allyn and Bacon .
   Google Scholar
• Tamrazov , PM . 1966 . Continuity of certain conformal invariants . Ukrain. Mat. Zh. , 18 ( 6 ) : 78 – 84 . (Russian)
   Google Scholar
• Ziemer , WP . 1969 . Extremal length and p-capacity . Michigan Math. J. , 16 : 43 – 51 .
   Web of Science ®Google Scholar