Finite distortion functions and Douglas-Dirichlet functionals

Shi, Qingtian

Czechoslovak Mathematical Journal, Tome 69 (2019) no. 1, pp. 183-195

Abstract (VO)

In this paper, we estimate the Douglas-Dirichlet functionals of harmonic mappings, namely Euclidean harmonic mapping and flat harmonic mapping, by using the extremal dilatation of finite distortion functions with given boundary value on the unit circle. In addition, $\bar {\partial }$-Dirichlet functionals of harmonic mappings are also investigated.

Détail

DOI : 10.21136/CMJ.2018.0238-17

Classification : 30C62, 30C70, 31A05
Keywords: Douglas-Dirichlet functional; $\rho $-harmonic mapping; finite distortion functions; extremal quasiconformal mapping; Dirichlet's principle

Bibliographie

[1] Astala, K., Iwaniec, T., Martin, G. J.: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton Mathematical Series 48, Princeton University Press, Princeton (2009). | DOI | MR | JFM

[2] Astala, K., Iwaniec, T., Martin, G. J., Onninen, J.: Extremal mappings of finite distortion. Proc. Lond. Math. Soc., III. Ser. 91 (2005), 655-702. | DOI | MR | JFM

[3] Chen, X., Qian, T.: Estimation of hyperbolically partial derivatives of $\rho$-harmonic quasiconformal mappings and its applications. Complex Var. Elliptic Equ. 60 (2015), 875-892. | DOI | MR | JFM

[4] Duren, P.: Harmonic Mappings in the Plane. Cambridge Tracts in Mathematics 156, Cambridge University Press, Cambridge (2004). | DOI | MR | JFM

[5] Feng, X.: Mappings of finite distortion and harmonic functions. Pure Appl. Math. 32 (2016), 119-126 Chinese. | DOI | JFM

[6] Feng, X., Tang, S.: A note on the $\rho$-Nitsche conjecture. Arch. Math. 107 (2016), 81-88. | DOI | MR | JFM

[7] Hencl, S., Koskela, P., Onninen, J.: A note on extremal mappings of finite distortion. Math. Res. Lett. 12 (2005), 231-237. | DOI | MR | JFM

[8] Kalaj, D., Mateljević, M.: Inner estimate and quasiconformal harmonic maps between smooth domains. J. Anal. Math. 100 (2006), 117-132. | DOI | MR | JFM

[9] Li, Z.: On the boundary value problem for harmonic maps of the Poincaré disc. Chin. Sci. Bull. 42 (1997), 2025-2045. | DOI | MR | JFM

[10] Mateljević, M.: Dirichlet's principle, distortion and related problems for harmonic mappings. Publ. Inst. Math., Nouv. Sér. 75(89) (2004), 147-171. | DOI | MR | JFM

[11] Mateljević, M.: Dirichlet's principle, uniqueness of harmonic maps and extremal QC mappings. Zbornik Radova (Beograd) 10(18) (2004), 41-91. | MR | JFM

[12] Qi, Y., Shi, Q.: Quasi-isometricity and equivalent moduli of continuity of planar $1/|\omega|^2$-harmonic mappings. Filomat 31 (2017), 335-345. | DOI | MR

[13] Reich, E.: On the variational principle of Gerstenhaber and Rauch. Ann. Acad. Sci. Fenn., Ser. A I, Math. 10 (1985), 469-475. | DOI | MR | JFM

[14] Reich, E.: Harmonic mappings and quasiconformal mappings. J. Anal. Math. 46 (1986), 239-245. | DOI | MR | JFM

[15] Schoen, R., Yau, T. S.: On univalent harmonic maps between surfaces. Invent. Math. 44 (1978), 265-278. | DOI | MR | JFM

[16] Shen, Y.: Quasiconformal mappings and harmonic functions. Adv. Math., Beijing 28 (1999), 347-357. | MR | JFM

[17] Shen, Y.: Extremal problems for quasiconformal mappings. J. Math. Anal. Appl. 247 (2000), 27-44. | DOI | MR | JFM

[18] Wei, H.: On the uniqueness problem of harmonic quasiconformal mappings. Proc. Am. Math. Soc. 124 (1996), 2337-2341. | DOI | MR | JFM

[19] Yao, G.: $\bar{\partial}$-energy integral and harmonic mappings. Proc. Am. Math. Soc. 131 (2003), 2271-2277. | DOI | MR | JFM

[20] Yao, G.: $\bar{\partial}$-energy integral and uniqueness of harmonic maps. Math. Nachr. 278 (2005), 1086-1096. | DOI | MR | JFM