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In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation

${\displaystyle {\partial w \over \partial {\overline {z}}}=\mu {\partial w \over \partial z}.}$

for w a complex distribution of the complex variable z in some open set U, with derivatives that are locally L, and where μ is a given complex function in L(U) of norm less than 1, called the Beltrami coefficient.

The Beltrami equation characterises quasiconformal mappings.

See also

• Measurable Riemann mapping theorem
• Isothermal coordinates

References

• Ahlfors, Lars V. (1966), Lectures on quasiconformal mappings, Van Nostrand
• Beltrami, Eugenio (1867), "Saggio di interpretazione della geometria non euclidea (Essay on the interpretation of noneuclidean geometry)" (PDF), Giornale di Mathematica (in Italian), 6, JFM 01.0275.02 English translation in Stillwell (1996)
• Douady, Adrien; Buff, X. (2000), Le théorème d'intégrabilité des structures presque complexes. , London Math. Soc. Lecture Note Ser., vol. 274, Cambridge Univ. Press, pp. 307–324
• Hubbard, John Hamal (2006), Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1, Matrix Editions, Ithaca, NY, ISBN 978-0-9715766-2-9, MR 2245223
• Imayoshi, Y.; Taniguchi, M. (1992), An Introduction to Teichmüller spaces, Springer-Verlag, ISBN 0-387-70088-9
• Iwaniec, Tadeusz; Martin, Gaven (2008), The Beltrami equation, Memoirs of the American Mathematical Society, vol. 191, ISBN 978-0-8218-4045-0, MR 2377904
• Morrey, Charles B. (1936), "On the solutions of quasi-linear elliptic partial differential equations.", Bulletin of the American Mathematical Society, 42 (5): 316, doi:10.1090/S0002-9904-1936-06297-X, ISSN 0002-9904, JFM 62.0565.02
• Morrey, Charles B. Jr. (1938), "On the Solutions of Quasi-Linear Elliptic Partial Differential Equations", Transactions of the American Mathematical Society, 43 (1), American Mathematical Society: 126–166, doi:10.2307/1989904, JSTOR 1989904, Zbl 0018.40501
• Papadopoulos, Athanase, ed. (2007), Handbook of Teichmüller theory. Vol. I, IRMA Lectures in Mathematics and Theoretical Physics, 11, European Mathematical Society (EMS), Zürich, doi:10.4171/029, ISBN 978-3-03719-029-6, MR2284826
• Papadopoulos, Athanase, ed. (2009), Handbook of Teichmüller theory. Vol. II, IRMA Lectures in Mathematics and Theoretical Physics, 13, European Mathematical Society (EMS), Zürich, doi:10.4171/055, ISBN 978-3-03719-055-5, MR2524085
• Stillwell, John (1996), Sources of hyperbolic geometry, History of Mathematics, vol. 10, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0529-9, MR 1402697

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