On quasiconformal self-mappings of the unit disk satisfying Poisson’s equation

Kalaj, David; Pavlović, Miroslav

doi:10.1090/S0002-9947-2011-05081-6

On quasiconformal self-mappings of the unit disk satisfying Poisson’s equation
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Trans. Amer. Math. Soc. 363 (2011), 4043-4061 Request permission

Abstract:

Let $\mathcal {QC}(K,g)$ be a family of $K$-quasiconformal mappings of the open unit disk onto itself satisfying the PDE $\Delta w =g$, $g\in C(\overline {\mathbb {U}})$, $w(0)=0$. It is proved that $\mathcal {QC}(K,g)$ is a uniformly Lipschitz family. Moreover, if $|g|_\infty$ is small enough, then the family is uniformly bi-Lipschitz. The estimations are asymptotically sharp as $K \to 1$ and $|g|_\infty \to 0$, so $w\in \mathcal {QC}(K,g)$ behaves almost like a rotation for sufficiently small $K$ and $|g|_\infty$.

References

Lars V. Ahlfors, Lectures on quasiconformal mappings, 2nd ed., University Lecture Series, vol. 38, American Mathematical Society, Providence, RI, 2006. With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard. MR 2241787, DOI 10.1090/ulect/038
G. D. Anderson and M. K. Vamanamurthy, Hölder continuity of quasiconformal mappings of the unit ball, Proc. Amer. Math. Soc. 104 (1988), no. 1, 227–230. MR 958072, DOI 10.1090/S0002-9939-1988-0958072-6
Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic function theory, Graduate Texts in Mathematics, vol. 137, Springer-Verlag, New York, 1992. MR 1184139, DOI 10.1007/b97238
Gustave Choquet, Sur un type de transformation analytique généralisant la représentation conforme et définie au moyen de fonctions harmoniques, Bull. Sci. Math. (2) 69 (1945), 156–165 (French). MR 16973
Richard Fehlmann and Matti Vuorinen, Mori’s theorem for $n$-dimensional quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 13 (1988), no. 1, 111–124. MR 975570, DOI 10.5186/aasfm.1988.1304
Paul R. Halmos, Measure Theory, D. Van Nostrand Co., Inc., New York, N. Y., 1950. MR 0033869
Juha Heinonen, Lectures on Lipschitz analysis, Report. University of Jyväskylä Department of Mathematics and Statistics, vol. 100, University of Jyväskylä, Jyväskylä, 2005. MR 2177410
Erhard Heinz, On one-to-one harmonic mappings, Pacific J. Math. 9 (1959), 101–105. MR 104933
Lars Hörmander, Notions of convexity, Progress in Mathematics, vol. 127, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1301332
David Kalaj, Quasiconformal and harmonic mappings between Jordan domains, Math. Z. 260 (2008), no. 2, 237–252. MR 2429610, DOI 10.1007/s00209-007-0270-9
David Kalaj, On harmonic quasiconformal self-mappings of the unit ball, Ann. Acad. Sci. Fenn. Math. 33 (2008), no. 1, 261–271. MR 2386850
David Kalaj and Miodrag Mateljević, Inner estimate and quasiconformal harmonic maps between smooth domains, J. Anal. Math. 100 (2006), 117–132. MR 2303306, DOI 10.1007/BF02916757
—, On certain nonlinear elliptic PDE and quasiconfomal mapps between Euclidean surfaces, Potential Analysis 34 (2011), no. 1, 13–22.
David Kalaj and Miroslav Pavlović, Boundary correspondence under quasiconformal harmonic diffeomorphisms of a half-plane, Ann. Acad. Sci. Fenn. Math. 30 (2005), no. 1, 159–165. MR 2140304
H. Kneser: Lösung der Aufgabe 41, Jahresber. Deutsch. Math.-Verein. 35 (1926), 123–124.
Miljan Knežević and Miodrag Mateljević, On the quasi-isometries of harmonic quasiconformal mappings, J. Math. Anal. Appl. 334 (2007), no. 1, 404–413. MR 2332565, DOI 10.1016/j.jmaa.2006.12.069
O. Martio, On harmonic quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I No. 425 (1968), 10. MR 0236382
M. Mateljević and M. Vuorinen, On harmonic quasiconformal quasi-isometries, J. Inequal. Appl. , posted on (2010), Art. ID 178732, 19. MR 2665497, DOI 10.1155/2010/178732
Akira Mori, On an absolute constant in the theory of quasi-conformal mappings, J. Math. Soc. Japan 8 (1956), 156–166. MR 79091, DOI 10.2969/jmsj/00820156
Dariusz Partyka and Ken-ichi Sakan, On bi-Lipschitz type inequalities for quasiconformal harmonic mappings, Ann. Acad. Sci. Fenn. Math. 32 (2007), no. 2, 579–594. MR 2337496
Miroslav Pavlović, Boundary correspondence under harmonic quasiconformal homeomorphisms of the unit disk, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 2, 365–372. MR 1922194
Songliang Qiu, On Mori’s theorem in quasiconformal theory, Acta Math. Sinica (N.S.) 13 (1997), no. 1, 35–44. A Chinese summary appears in Acta Math. Sinica 40 (1997), no. 2, 319. MR 1465533, DOI 10.1007/BF02560522
T. Radó, Aufgabe 41. (Gestellt in Jahresbericht D. M. V. 35, 49) Lösung von H. Kneser, Jahresbericht D. M. V. 35, 123–124 ((1926)) (German).
Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
Erik Talvila, Necessary and sufficient conditions for differentiating under the integral sign, Amer. Math. Monthly 108 (2001), no. 6, 544–548. MR 1840661, DOI 10.2307/2695709
Luen-Fai Tam and Tom Y.-H. Wan, Harmonic diffeomorphisms into Cartan-Hadamard surfaces with prescribed Hopf differentials, Comm. Anal. Geom. 2 (1994), no. 4, 593–625. MR 1336897, DOI 10.4310/CAG.1994.v2.n4.a5
Luen-Fai Tam and Tom Y. H. Wan, Quasi-conformal harmonic diffeomorphism and the universal Teichmüller space, J. Differential Geom. 42 (1995), no. 2, 368–410. MR 1366549
Luen-Fai Tam and Tom Y.-H. Wan, On quasiconformal harmonic maps, Pacific J. Math. 182 (1998), no. 2, 359–383. MR 1609587, DOI 10.2140/pjm.1998.182.359
Vassilij S. Vladimirov, Equazioni della fisica matematica, Edizioni Mir, Moscow, 1987 (Italian). Translated from the fourth Russian edition by Ernest Kozlov. MR 1018346
Tom Yau-Heng Wan, Constant mean curvature surface, harmonic maps, and universal Teichmüller space, J. Differential Geom. 35 (1992), no. 3, 643–657. MR 1163452
Zhong Li and Gui Zhen Cui, A note on Mori’s theorem of $K$-quasiconformal mappings, Acta Math. Sinica (N.S.) 9 (1993), no. 1, 55–62. A Chinese summary appears in Acta Math. Sinica 37 (1994), no. 2, 287. MR 1235641, DOI 10.1007/BF02559983