On Functions with a Conjugate
Annales de l'Institut Fourier, Volume 65 (2015) no. 1, pp. 277-314.

Harmonic functions of two variables are exactly those that admit a conjugate, namely a function whose gradient has the same length and is everywhere orthogonal to the gradient of the original function. We show that there are also partial differential equations controlling the functions of three variables that admit a conjugate.

Les fonctions harmoniques en deux variables sont exactement celles qui admettent une fonction conjuguée, à savoir une fonction dont le gradient a la même longueur et est partout orthogonal au gradient de la fonction d’origine. Nous montrons qu’il existe des équations aux dérivées partielles qui contrôlent également les fonctions de trois variables qui admettent une fonction conjuguée.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.2931
Classification: 53A30
Keywords: conjugate function, conformal invariant, partial differential inequality, partial differential equation, 3-harmonic function, conformal Killing field
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Baird, Paul; Eastwood, Michael. On Functions with a Conjugate. Annales de l'Institut Fourier, Volume 65 (2015) no. 1, pp. 277-314. doi : 10.5802/aif.2931. https://aif.centre-mersenne.org/articles/10.5802/aif.2931/

[1] Ababou, Rachel; Baird, Paul; Brossard, Jean Polynômes semi-conformes et morphismes harmoniques, Math. Z., Tome 231 (1999) no. 3, pp. 589-604 | Article | MR: 1704994 | Zbl: 0963.53036

[2] Baird, P.; Eastwood, M.G. Singularities of semiconformal mappings, RIMS Kokyuroku, Tome 1610 (2008), pp. 1-10

[3] Baird, Paul; Eastwood, Michael Conjugate functions and semiconformal mappings, Differential geometry and its applications, Matfyzpress, Prague, 2005, pp. 479-486 | MR: 2268958 | Zbl: 1107.53024

[4] Baird, Paul; Eastwood, Michael CR geometry and conformal foliations, Ann. Global Anal. Geom., Tome 44 (2013) no. 1, pp. 73-90 | Article | MR: 3055804 | Zbl: 1268.53060

[5] Baird, Paul; Pantilie, Radu Harmonic morphisms on heaven spaces, Bull. Lond. Math. Soc., Tome 41 (2009) no. 2, pp. 198-204 | Article | MR: 2496497 | Zbl: 1173.53027

[6] Baird, Paul; Wood, John C. Harmonic morphisms between Riemannian manifolds, London Mathematical Society Monographs. New Series, Tome 29, The Clarendon Press, Oxford University Press, Oxford, 2003, xvi+520 pages | Article | MR: 2044031 | Zbl: 1055.53049

[7] Bojarski, B.; Iwaniec, T. p-harmonic equation and quasiregular mappings, Partial differential equations (Warsaw, 1984) (Banach Center Publ.) Tome 19, PWN, Warsaw, 1987, pp. 25-38 | MR: 1055157 | Zbl: 0659.35035

[8] Eastwood, Michael Higher symmetries of the Laplacian, Ann. of Math. (2), Tome 161 (2005) no. 3, pp. 1645-1665 | Article | MR: 2180410 | Zbl: 1091.53020

[9] Eastwood, Michael G.; Graham, C. Robin Invariants of conformal densities, Duke Math. J., Tome 63 (1991) no. 3, pp. 633-671 | Article | MR: 1121149 | Zbl: 0745.53007

[10] Hardt, Robert; Lin, Fang-Hua Mappings minimizing the L p norm of the gradient, Comm. Pure Appl. Math., Tome 40 (1987) no. 5, pp. 555-588 | Article | MR: 896767 | Zbl: 0646.49007

[11] Loubeau, E. On p-harmonic morphisms, Differential Geom. Appl., Tome 12 (2000) no. 3, pp. 219-229 | Article | MR: 1764330 | Zbl: 0966.58009

[12] Nurowski, Paweł Construction of conjugate functions, Ann. Global Anal. Geom., Tome 37 (2010) no. 4, pp. 321-326 | Article | MR: 2601492 | Zbl: 1190.53045

[13] Penrose, Roger; Rindler, Wolfgang Spinors and space-time. Vol. 1, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1984, x+458 pages (Two-spinor calculus and relativistic fields) | Article | MR: 776784 | Zbl: 0538.53024

[14] Szekeres, Peter Conformal tensors., Proc. R. Soc. Lond., Ser. A, Tome 304 (1968), pp. 113-122 | Article | Zbl: 0159.23903

[15] Weyl, Hermann The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, N.J., 1939, xii+302 pages | MR: 1488158 | Zbl: 1024.20502

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