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The Robin function and conformal welding – A new proof of the existence

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Veröffentlicht/Copyright: 15. Februar 2018
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Abstract

Green’s function of the mixed boundary value problem for harmonic functions is sometimes named the Robin function R(z,ζ) after the French mathematical physicist Gustave Robin (1855–1897). The aim of this paper is to provide a new proof of the existence of the Robin function for planar n-fold connected domains using a special version of the well-known Koebe’s uniformization theorem and a conformal mapping which is closely related to the Robin function in the simply connected case.

Keywords: Robin function
MSC 2010: 35J

Dedicated to Reiner Kühnau on the occasion of his 80th birthday


Acknowledgements

The author wishes to express his thanks to Reiner Kühnau for some hints to the references and Markus Köbis for drawing the figures.

References

[1] M. G. Arsove and G. Johnson, Jr., A conformal mapping technique for infinitely connected regions, Mem. Amer. Math. Soc. 91 (1970), 1–56. 10.1090/memo/0091Suche in Google Scholar

[2] H. Begehr and T. Vaitekhovich, Some harmonic Robin functions in the complex plane, Adv. Pure Appl. Math. 1 (2010), no. 1, 19–34. 10.1515/apam.2010.003Suche in Google Scholar

[3] S. Bergman and M. Schiffer, Kernel Functions and Elliptic Differential Equations in Mathematical Physics, Academic Press, New York, 1953. Suche in Google Scholar

[4] M. Brelot and G. Choquet, Espaces et lignes de Green, Ann. Inst. Fourier Grenoble 3 (1951), 199–263. 10.5802/aif.38Suche in Google Scholar

[5] B. Dittmar, Characterization of the Robin function by extremal problems, Georgian Math. J. 21 (2014), no. 3, 267–271. 10.1515/gmj-2014-0027Suche in Google Scholar

[6] B. Dittmar and M. Hantke, The Robin function and its eigenvalues, Georgian Math. J. 14 (2007), no. 3, 403–417. 10.1515/GMJ.2007.403Suche in Google Scholar

[7] B. Dittmar and R. Kühnau, Zur Konstruktion der Eigenfunktionen Stekloffscher Eigenwertaufgaben, Z. Angew. Math. Phys. 51 (2000), no. 5, 806–819. 10.1007/PL00001520Suche in Google Scholar

[8] B. Dittmar and A. Y. Solynin, Distortion of the hyperbolic Robin capacity under a conformal mapping, and extremal configurations (in Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 263 (2000), 49–69; translation in J. Math. Sci. (New York) 110 (2002), no. 6, 3058–3069. Suche in Google Scholar

[9] V. N. Dubinin, On quadratic forms generated by Green and Robin functions (in Russian), Mat. Sb. 200 (2009), no. 10, 25–38; translation in Sb. Math. 200 (2009), no. 9-10, 1439–1452. Suche in Google Scholar

[10] V. N. Dubinin and M. Vuorinen, Robin functions and distortion theorems for regular mappings, Math. Nachr. 283 (2010), no. 11, 1589–1602. 10.1002/mana.200710143Suche in Google Scholar

[11] P. Duren, Robin capacity, Computational Methods and Function Theory (Nicosia 1997), World Sci. Ser. Approx. Decompos. 11, World Scientific Publisher, River Edge (1999), 177–190. 10.1142/9789812833044_0013Suche in Google Scholar

[12] P. Duren and R. Kühnau, Elliptic capacity and its distortion under conformal mapping, J. Anal. Math. 89 (2003), 317–335. 10.1007/BF02893086Suche in Google Scholar

[13] P. Duren and J. Pfaltzgraff, Robin capacity and extremal length, J. Math. Anal. Appl. 179 (1993), no. 1, 110–119. 10.1006/jmaa.1993.1338Suche in Google Scholar

[14] P. Duren and J. Pfaltzgraff, Hyperbolic capacity and its distortion under conformal mapping, J. Anal. Math. 78 (1999), 205–218. 10.1007/BF02791134Suche in Google Scholar

[15] P. Duren, J. Pfaltzgraff and E. Thurman, Physical interpretation and further properties of Robin capacity (in Russian), Algebra i Analiz 9 (1997), no. 3, 211–219; translation in St. Petersburg Math. J. 9 (1998), no. 3, 607–614. Suche in Google Scholar

[16] P. Duren and M. M. Schiffer, Robin functions and distortion of capacity under conformal mapping, Complex Variables Theory Appl. 21 (1993), no. 3–4, 189–196. 10.1080/17476939308814628Suche in Google Scholar

[17] P. L. Duren and M. M. Schiffer, Robin functions and energy functionals of multiply connected domains, Pacific J. Math. 148 (1991), no. 2, 251–273. 10.2140/pjm.1991.148.251Suche in Google Scholar

[18] G. M. Golusin, Geometrische Funktionentheorie, Hochschulb. Math. 31, VEB Deutscher Verlag der Wissenschaften, Berlin, 1957. Suche in Google Scholar

[19] H. Grötzsch, Über die Verzerrung bei schlichter konformer Abbildung mehrfach zusammenhängender Bereiche II, Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math. Naturwiss. Kl 81 (1929), 217–221. Suche in Google Scholar

[20] H. Grunsky, Lectures on Theory of Functions in Multiply Connected Domains, Vandenhoeck & Ruprecht, Göttingen, 1978. Suche in Google Scholar

[21] K. Gustafson and T. Abe, The third boundary condition—was it Robin’s?, Math. Intelligencer 20 (1998), no. 1, 63–71. 10.1007/BF03024402Suche in Google Scholar

[22] K. Gustafson and T. Abe, (Victor) Gustave Robin: 1855–1897, Math. Intelligencer 20 (1998), no. 2, 47–53. 10.1007/BF03025298Suche in Google Scholar

[23] P. Koebe, Über die konforme Abbildung mehrfach zusammenhängender Bereiche, Deutsche Math. Ver. 19 (1910), 339–348. Suche in Google Scholar

[24] P. Koebe, Abhandlungen zur Theorie der konformen Abbildung. IV. Abbildung mehrfach zusammenhängender schlichter Bereiche auf Schlitzbereiche, Acta Math. 41 (1916), no. 1, 305–344. 10.1007/BF02422949Suche in Google Scholar

[25] P. Koebe, Allgemeine Theorie der Riemannschen Mannigfaltigkeiten (Konforme Abbildung und Uniformisierung), Acta Math. 50 (1927), no. 1, 27–157. 10.1007/BF02421322Suche in Google Scholar

[26] A. Mohammed and M. W. Wong, Solutions of the Riemann–Hilbert–Poincaré problem and the Robin problem for the inhomogeneous Cauchy–Riemann equation, Proc. Roy. Soc. Edinburgh Sect. A 139 (2009), no. 1, 157–181. 10.1017/S0308210507000108Suche in Google Scholar

[27] S. Nasyrov, Robin capacity and lift of infinitely thin airfoils, Complex Var. Theory Appl. 47 (2002), no. 2, 93–107. 10.1080/02781070290010850Suche in Google Scholar

[28] R. Nevanlinna, Eindeutige analytische Funktionen, 2te Auflage, Grundlehren Math. Wiss. 46, Springer, Berlin, 1953. 10.1007/978-3-662-06842-7Suche in Google Scholar

[29] M. D. O’Neill and R. E. Thurman, Extremal domains for Robin capacity, Complex Variables Theory Appl. 41 (2000), no. 1, 91–109. 10.1080/17476930008815239Suche in Google Scholar

[30] C. Pommerenke, On the logarithmic capacity and conformal mapping, Duke Math. J. 35 (1968), 321–325. 10.1215/S0012-7094-68-03531-XSuche in Google Scholar

[31] C. Pommerenke, Univalent Functions, Studia Math. 25, Vandenhoeck & Ruprecht, Göttingen, 1975. Suche in Google Scholar

[32] A. C. Schaeffer and D. C. Spencer, Coefficient Regions for Schlicht Functions, Amer. Math. Soc. Colloq. Publ. 35, American Mathematical Society, New York, 1950. 10.2307/3611490Suche in Google Scholar

[33] M. Schiffer, Hadamard’s formula and variation of domain-functions, Amer. J. Math. 68 (1946), 417–448. 10.2307/2371824Suche in Google Scholar

[34] A. Y. Solynin, A note on equilibrium points of Green’s function, Proc. Amer. Math. Soc. 136 (2008), no. 3, 1019–1021. 10.1090/S0002-9939-07-09156-3Suche in Google Scholar

[35] M. Stiemer, A representation formula for the Robin function, Complex Var. Theory Appl. 48 (2003), no. 5, 417–427. 10.1080/0278107031000094666Suche in Google Scholar

[36] M. Stiemer, Extremal point methods for Robin capacity, Comput. Methods Funct. Theory 4 (2004), no. 2, 475–496. 10.1007/BF03321082Suche in Google Scholar

[37] R. E. Thurman, Maximal capacity, Robin capacity, and minimum energy, Indiana Univ. Math. J. 46 (1997), no. 2, 621–636. 10.1512/iumj.1997.46.1336Suche in Google Scholar

[38] M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959. Suche in Google Scholar

[39] A. Vasil’ev, Robin’s modulus in a Hele–Shaw problem, Complex Var. Theory Appl. 49 (2004), no. 7–9, 663–672. 10.1080/02781070410001732188Suche in Google Scholar

Received: 2015-04-30
Revised: 2016-05-23
Accepted: 2016-06-22
Published Online: 2018-02-15
Published in Print: 2020-03-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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