On Functions with a Conjugate
[Sur les fonctions qui admettent une fonction conjuguée]
Baird, Paul ; Eastwood, Michael
Annales de l'Institut Fourier, Tome 65 (2015), p. 277-314 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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.

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.

Publié le : 2015-01-01
DOI : https://doi.org/10.5802/aif.2931
Classification:  53A30
Mots clés: fonction conjuguée, invariant conforme, inégalité aux dérivées partielles, équation aux dérivées partielles, fonction 3-harmonique, champ de Killing conforme 
@article{AIF_2015__65_1_277_0,
     author = {Baird, Paul and Eastwood, Michael},
     title = {On Functions with a Conjugate},
     journal = {Annales de l'Institut Fourier},
     volume = {65},
     year = {2015},
     pages = {277-314},
     doi = {10.5802/aif.2931},
     zbl = {06496540},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2015__65_1_277_0}
}

Baird, Paul; Eastwood, Michael. On Functions with a Conjugate. Annales de l'Institut Fourier, Tome 65 (2015) pp. 277-314. doi : 10.5802/aif.2931. http://gdmltest.u-ga.fr/item/AIF_2015__65_1_277_0/

[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, The Clarendon Press, Oxford University Press, Oxford, London Mathematical Society Monographs. New Series, Tome 29 (2003), pp. xvi+520 | Article | MR 2044031 | Zbl 1055.53049

[7] Bojarski, B.; Iwaniec, T. p-harmonic equation and quasiregular mappings, Partial differential equations (Warsaw, 1984), PWN, Warsaw (Banach Center Publ.) Tome 19 (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 University Press, Cambridge, Cambridge Monographs on Mathematical Physics (1984), pp. x+458 (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), pp. xii+302 | MR 1488158 | Zbl 1024.20502