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References
Ahlfors, Lars V., Lectures on Quasiconformal Mappings. Princeton: Van Nostrand 1966.
Ahlfors, Lars V., & Lipman Bers, Riemann's mapping theorem for variable metrics. Ann. Math. 72, 385–404 (1960).
Beurling, A., & Lars V. Ahlfors, The boundary correspondence under quasiconformal mappings. Acta Math. 96, 125–142 (1956).
Duren, P.L., & M. Schiffer, The theory of the second variation in extremum problems for univalent functions. Journ. d'Anal. Math. 10, 193–252 (1963).
Garabedian, P.R., & M. Schiffer, Convexity of domain functionals. Journ. d'Anal. Math. 2, 281–368 (1952/53).
Gehring, F.W., Rings and quasiconformal mappings in space. Trans. Amer. Math. Soc. 103, 353–393 (1962).
Golusin, G.M., Geometrische Funktionentheorie. Berlin: VEB Deutscher Verlag der Wissenschaften 1957.
Hadamard, J., Mémoire sur le problème d'analyse relatif à l'équilibre des plaques élastiques encastrées. Acad. Sci. Paris, Mémoires des savants étrangers 33, 1–128 (1908) or Oeuvres 2, 515–629. Centre National de la Recherche Scientifique. Paris, 1968.
Hadamard, J., Leçons sur le Calcul des Variations. Paris: Hermann 1910.
Kazdan, J.L., A boundary value problem arising in the theory of univalent functions. Journ. Math. Mech. 13, 283–303 (1964).
Kelingos, J.A., Boundary correspondence under quasiconformal mappings. Mich. Math. Journ. 13, 235–249 (1966).
Kelingos, J.A., On the maximal dilatation of quasiconformal extensions. Ann. Acad. Sci. Fenn. Ser. AI 478, 8 pp. (1970).
Lehto, O., & K.I. Virtanen, Quasikonforme Abbildungen. Berlin-Heidelberg-New York: Springer 1965.
Lehto, O., K.I. Virtanen, & J. Väisälä, Contributions to the distortion theory of quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. AI 273, 14 pp. (1959).
Mori, A., On quasiconformality and pseudoanalyticity. Trans. Amer. Math. Soc. 84, 56–77 (1957).
Nehari, A., Conformal Mapping. New York: Mc-Graw Hill 1952.
Sario, L., & M. Nakai, Classification Theory of Riemann Surfaces. Berlin-Heidelberg-New York: Springer 1970.
Schiffer, M., Variation of the Green's function and theory of the p-valued functions. Amer. Journ. Math. 65, 341–360 (1943).
Schiffer, M., Hadamard's formula and variation of domain functions. Amer. Journ. Math. 68, 417–448 (1946).
Schiffer, M., Some recent developments in the theory of conformal mapping. Appendix to Courant, R., Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces. New York: Interscience Publishers 1950.
Schiffer, M., Variational methods in the theory of conformal mapping. Proc. Intl. Cong. Math. 233–240 (1950).
Schiffer, M., Applications of variational methods in the theory of conformal mapping. Proc. Symp. Appl. Math., Amer. Math. Soc., vol. VIII, 93–113. New York: Mc-Graw Hill 1958.
Schiffer, M., Extremum problems and variational methods in conformal mapping. Proc. Intl. Cong. Math. 211–231 (1958).
Warschawski, S.E., On Hadamard's variation formula for Green's function. Journ. Math. Mech. 9, 497–511 (1960).
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Communicated by M. M. Schiffer
This work was partially supported by the National Science Foundation and by the Eidgenössische Technische Hochschule, Zürich. The results form the author's doctoral thesis at the University of Minnesota. The author especially wishes to thank her thesis advisor, Prof. Edgar Reich, for his help and encouragement during the preparation of that thesis.
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Sontag, A. Variation of the Green's function due to quasiconformal distortion of the region. Arch. Rational Mech. Anal. 59, 257–280 (1975). https://doi.org/10.1007/BF00251603
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DOI: https://doi.org/10.1007/BF00251603