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Quasiconformal Harmonic Mappings Between the Unit Ball and a Spatial Domain with C1,α Boundary

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Abstract

We prove the following. If f is a harmonic quasiconformal mapping between the unit ball in \(\mathbb {R}^{n}\) and a spatial domain with C1,α boundary, then f is Lipschitz continuous in B. This generalizes some known results for n = 2 and improves some others in higher dimensional case.

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References

  1. Arsenović, M., Božin, V., Manojlović, V.: Moduli of continuity of harmonic quasiregular mappings in Bn. Potential Anal 34, 283–291 (2011)

    Article  MathSciNet  Google Scholar 

  2. Astala, K., Manojlović, V.: On Pavlovic theorem in space. Potential Anal. 43(3), 361–370 (2015)

    Article  MathSciNet  Google Scholar 

  3. Axler, S., Bourdon, P., Ramey, W.: Harmonic Function Theory. Springer-Verlag, New York (2000)

    MATH  Google Scholar 

  4. Božin, V., Mateljević, M.: Quasiconformal and HQC mappings between Lyapunov Jordan domains. Ann. Sc. Norm. Super. Pisa, Cl. Sci. pp. 23(5). https://doi.org/10.2422/2036-2145.201708∖_013

  5. Fehlmann, R., Vuorinen, M.: Mori’s theorem for n-dimensional quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. A I Math. 13(1), 111–124 (1988)

    Article  MathSciNet  Google Scholar 

  6. Gehring, F.W., Martio, O.: Lipschitz classes and quasiconformal mappings. Ann. Acad. Sci. Fenn., Ser. A I, Math. 10, 203–219 (1985)

    Article  MathSciNet  Google Scholar 

  7. Goluzin, G.M.: Geometric theory of functions of a complex variable. Translations of Mathematical Monographs. Vol. 26. Providence, R. I.:American Mathematical Society (AMS). vi, 676 pp. (1969)

  8. Kalaj, D.: Quasiconformal harmonic mapping between Jordan domains. Math. Z. 260(2), 237–252 (2008)

    Article  MathSciNet  Google Scholar 

  9. Kalaj, D.: Quasiconformal harmonic mappings between Dini’s smooth Jordan domains. Pac. J. Math. 276, 213–228 (2015)

    Article  MathSciNet  Google Scholar 

  10. Kalaj, D.: A priori estimate of gradient of a solution to certain differential inequality and quasiconformal mappings. J. d’Analyse Math. 119, 63–88 (2013)

    Article  MathSciNet  Google Scholar 

  11. Kalaj, D.: On boundary correspondences under quasiconformal harmonic mappings between smooth Jordan domains. Math. Nachr. 285(2-3), 283–294 (2012)

    Article  MathSciNet  Google Scholar 

  12. Kalaj, D.: Harmonic mappings and distance function. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 10(3), 669–681 (2011)

    MathSciNet  MATH  Google Scholar 

  13. Kalaj, D.: Harmonic quasiconformal mappings betwen C1 smooth Jordan domains. arXiv:2003.03665. To appear in Revista Matemática Iberoamericana (2020)

  14. Kalaj, D., Lamel, B.: Minimisers and Kellogg’s theorem. Math. Ann. 377(3), 1643–1672 (2020)

    Article  MathSciNet  Google Scholar 

  15. Kalaj, D., Mateljević, M.: (K, K’)-quasiconformal harmonic mappings. Potential Anal. 36(1), 117–135 (2012)

    Article  MathSciNet  Google Scholar 

  16. Kalaj, D., Pavlović, M.: Boundary correspondence under quasiconformal harmonic diffeomorphisms of a half-plane. Ann. Acad. Sci. Fenn., Math. 30(1), 159–165 (2005)

    MathSciNet  MATH  Google Scholar 

  17. Kalaj, D., Zlatičanin, A.: Quasiconformal mappings with controlled Laplacian and Hölder continuity. Ann. Acad. Sci. Fenn., Math. 44(2), 797–803 (2019)

    Article  MathSciNet  Google Scholar 

  18. Manojlović, V.: Bi-Lipschicity of quasiconformal harmonic mappings in the plane. Filomat 23(1), 85–89 (2009)

    Article  MathSciNet  Google Scholar 

  19. Martio, O.: On harmonic quasiconformal mappings. Ann. Acad. Sci. Fenn., Ser. A I 425, 3–10 (1968)

    MathSciNet  MATH  Google Scholar 

  20. Martio, O., Näkki, R.: Hölder continuity and quasiconformal mappings. J. Lond. Math. Soc. (2) 44(2), 339–350 (1991)

    Article  Google Scholar 

  21. Mateljević, M., Vuorinen, M.: On harmonic quasiconformal quasi-isometries. J. Inequal. Appl 2010, 19 (2010). Article ID 178732

    Article  MathSciNet  Google Scholar 

  22. Nitsche, J.C.C.: The boundary behavior of minimal surfaces. Kellogg’s theorem and branch points on the boundary. Invent. Math. 8, 313–333 (1969)

    Article  MathSciNet  Google Scholar 

  23. Partyka, D., Sakan, K.: On bi-Lipschitz type inequalities for quasiconformal harmonic mappings. Ann. Acad. Sci. Fenn. Math. 32, 579–594 (2007)

    MathSciNet  MATH  Google Scholar 

  24. Partyka, D., Sakan, K.I., Zhu, J.-F.: Quasiconformal harmonic mappings with the convex holomorphic part. Ann. Acad. Sci. Fenn., Math. 43(1), 401–418 (2018). erratum ibid. 43, No. 2, 1085–1086 (2018)

    Article  MathSciNet  Google Scholar 

  25. Pavlović, M.: Lipschitz conditions on the modulus of a harmonic function. Rev. Mat. Iberoam. 23(3), 831–845 (2007)

    Article  MathSciNet  Google Scholar 

  26. Pavlović, M.: Boundary correspondence under harmonic quasiconformal homeomorfisms of the unit disc. Ann. Acad. Sci. Fenn. 27, 365–372 (2002)

    MATH  Google Scholar 

  27. Väisälä, J.: Lectures on n-dimensional Quasiconformal Mappings Lecture notes Math., vol. 229. Srpinger-Verlag, Berlin-New York (1971)

    Book  Google Scholar 

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Acknowledgements

We are grateful to the anonymous referee for a number of corrections that have made this paper better.

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  1. Faculty of Natural Sciences and Mathematics, University of Montenegro, Cetinjski put b.b., 81000, Podgorica, Montenegro

    Anton Gjokaj & David Kalaj

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  1. Anton Gjokaj
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  2. David Kalaj
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Correspondence to David Kalaj.

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Gjokaj, A., Kalaj, D. Quasiconformal Harmonic Mappings Between the Unit Ball and a Spatial Domain with C1,α Boundary. Potential Anal 57, 367–377 (2022). https://doi.org/10.1007/s11118-021-09919-y

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  • DOI: https://doi.org/10.1007/s11118-021-09919-y

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