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Quaternionic Monge-Ampère equations. (English) Zbl 1058.32028
Das Hauptergebnis dieser Arbeit ist der Satz über die Existenz der eindeutigen Lösung des Dirichletschen Problems für die quaternionische Gleichung von Monge-Ampère in der Form \[ \text{det}\Biggl({\partial^2 u\over\partial q_i\partial\overline q_j}\Biggr)= f,\quad\text{in }\Omega \] in einem quaternionischen, streng pseudokonvexen Gebiet \(\Omega\subset\mathbb H^n\).

32W20 Complex Monge-Ampère operators
15B33 Matrices over special rings (quaternions, finite fields, etc.)
35J60 Nonlinear elliptic equations
32U05 Plurisubharmonic functions and generalizations
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[1] Aleksandrov, A.D. Dirichlet’s problem for the equation Det atDirichlet’s problem for the equation Det ||z ij ||=(z 1,...,z n ,z, x 1,...,x n ), I, (Russian),Vestnik Leningrad, Univ. Ser. Math. Meh. Astr.,13(1), 5–24, (1958). · Zbl 0114.30202
[2] Alesker, S., Non-commutative linear algebra and plurisubharmonic functions of quaternionic variables,Bull. Sci. Math.,127, 1–35, (also math. CV/0104209) (2003). · Zbl 1033.15013 · doi:10.1016/S0007-4497(02)00004-0
[3] Alesker, S. A topological obstruction to existence of quaternionic Plücker map, math. AT/0111028. · Zbl 1063.55016
[4] Arnold, V. Polymathematics: is a mathematics a single science or a set of arts? inMathematics: Frontiers and Perspectives, Arnold, V., Atiyah, M., Lax, P., and Mazur, B., Eds., (2000). · Zbl 0980.00004
[5] Artin, E.Geometric Algebra, Interscience Publishers, Inc., New York-London, (1957). · Zbl 0077.02101
[6] Aslaksen, H. Quaternionic determinants,Math. Intelligencer,18(3), 57–65, (1996). · Zbl 0881.15007 · doi:10.1007/BF03024312
[7] Aubin, T. Equations du type de Monge-Ampère sur les variétés kähleriennes compactes,C.R. Acad. Sci. Paris,283, 119–121, (1976). · Zbl 0333.53040
[8] Aubin, T. Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der Mathematischen Wissenschaften,Fundamental Principles of Mathematical Sciences,252, Springer-Verlag, New York, (1982). · Zbl 0512.53044
[9] Baird, P. and Wood, J.C. Harmonic morphisms between Riemannian manifolds, book in preparation. · Zbl 1055.53049
[10] Bedford, E. and Taylor, B.A. The Dirichlet problem for a complex Monge-Ampère equation,Invent. Math.,37(1), 1–44, (1976). · Zbl 0325.31013 · doi:10.1007/BF01418826
[11] Besse, A.L. Einstein manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete (3),Results in Mathematics and Related Areas (3),10, Springer-Verlag, Berlin, (1987). · Zbl 0613.53001
[12] Brackx, F., Delanghe, R., and Sommen, F. Clifford analysis,Research Notes in Mathematics,76, Pitman, (Advanced Publishing Program), Boston, MA, (1982).
[13] Bremermann, H.J. On a generalized Dirichlet problem for plurisubharmonic functions and pseudoconvex domains, Characterization of Šilov boundaries,Trans. Am. Math. Soc.,91, 246–276, (1959). · Zbl 0091.07501
[14] Caffarelli, L., Nirenberg, L., and Spruck, J. The Dirichlet problem for nonlinear second-order elliptic equations, I, Monge-Ampère equation,Comm. Pure Appl. Math.,37(3), 369–402, (1984). · Zbl 0598.35047 · doi:10.1002/cpa.3160370306
[15] Caffarelli, L., Kohn, J.J., Nirenberg, L., and Spruck, J. The Dirichlet problem for nonlinear second-order elliptic equations, II, Complex Monge-Ampère, and uniformly elliptic, equations,Comm. Pure Appl. Math.,38(2), 209–252, (1985). · Zbl 0598.35048 · doi:10.1002/cpa.3160380206
[16] Cheng, S.Y. and Yau, S.-T. On the regularity of the Monge-Ampère equation det( u/ i x j ) =F(x, u),Comm. Pure Appl. Math.,30(1), 41–68, (1977). · Zbl 0347.35019 · doi:10.1002/cpa.3160300104
[17] Cheng, S.Y. and Yau, S.-T. The real Monge-Ampére equation and affine flat structures,Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. 1, 2, 3, (Beijing, 1980), Science Press, Beijing, 339–370, (1982).
[18] Chern, S.S., Levine, H.I., and Nirenberg, L. Intrinsic norms on a complex manifold,1969 Global Analysis, (papers in Honor of K. Kodaira), 119–139, University Tokyo Press, Tokyo. · Zbl 0202.11603
[19] Demailly, J.-P. Complex analytic and differential geometry, book in preparation.
[20] Dieudonné, J. Les déterminants sur un corps non commutatif, (French),Bull. Soc. Math. France,71, 27–45, (1943). · Zbl 0028.33904
[21] Evans, L.C. Partial differential equations and Monge-Kantorovich mass transfer, Current developments in mathematics, 1997, (Cambridge, MA), 65–126, Int. Press, Boston, MA, (1999).
[22] Fueter, R. Die Funktionentheotie der Differentialgleichungen \(\Delta\)u=0 and \(\Delta\)\(\Delta\)u=0 mit vier reelen Variablen,Comment. Math. Helv.,7, 307–330, (1935). · Zbl 0012.01704 · doi:10.1007/BF01292723
[23] Fueter, R. Über die analytische Darstellung der regulären Functionen einer Quaternionenvariablen,Comment. Math. Helv.,8, 371–378, (1936). · Zbl 0014.16702 · doi:10.1007/BF01199562
[24] Fuglede, B. Harmonic morphisms between Riemannian manifolds,Ann. Inst. Fourier, (Grenoble),28(2, vi), 107–144, (1978). · Zbl 0339.53026 · doi:10.5802/aif.691
[25] Gelfand, I.M. and Shilov, G.E. Generalized functions. Vol. 1, Properties and operations. Translated from the Russian by Eugene Saletan, Academic Press, Harcourt Brace Jovanovich, Publishers, 1964, New York-London, (1977).
[26] Gelfand, I.M., Graev, M.I., and Vilenkin, N.Ya. Generalized functions, Vol. 5, Integral geometry and representation theory, Translated from the Russian by Eugene Saletan. Academic Press, Harcourt Brace Jovanovich, Publishers, 1966, New York-London, (1977). · Zbl 0144.17202
[27] Gelfand, I. and Retakh, V. Determinants of matrices over noncommutative rings, (Russian),Funktsional. Anal. i Prilozhen,25(2), 13–25, 96, (1991), translation inFunct. Anal. Appl.,25(2), 91–102, (1991).
[28] Gelfand, I. and Retakh, V. Quansideterminants. I,Selecta Math. (N.S.),3(4), 517–546, (1997). · Zbl 0919.16011 · doi:10.1007/s000290050019
[29] Gelfand, I., Retakh, V., and Wilson, R.L. Quaternionic quasideterminants and determinants, math. QA/0206211. · Zbl 1037.15006
[30] Gelfand, I., Gelfand, S., Retakh, V., and Wilson, R.L. Quasideterminants, math. QA/0208146.
[31] Gilbarg, D. and Trudinger, N.S. Elliptic partial differential equations of second order, 2nd ed.,Grundlehren der Mathematischen Wissenschaften,224, Springer-Verlag, Berlin, (1983). · Zbl 0562.35001
[32] Goffman, C. and Serrin, J. Sublinear functions of measures and variational integrals,Duke Math. J.,31, 159–178, (1964). · Zbl 0123.09804 · doi:10.1215/S0012-7094-64-03115-1
[33] Gromov, M. Private communication.
[34] Gürsey, F. and Tze, H.C. Complex and quaternionic analyticity in chiral and gauge theories, I,Ann. Physics,128(1), 29–130, (1980). · Zbl 0457.30039 · doi:10.1016/0003-4916(80)90056-1
[35] Hamilton, W.R. Elements of quaternions, Vols. I, II, Charles, Jasper Joly, Ed., Chelsea Publishing Co., New York, (1969).
[36] Henkin, G. Private communication.
[37] Hörmander, L. An introduction to complex analysis in several variables, 3rd ed.,North-Holland Mathematical Library,7, North-Holland Publishing Co., Amsterdam, (1990). · Zbl 0685.32001
[38] Hörmander, L. Notions of convexity,Progress in Mathematics,127, Birkhäuser, Boston, Inc., Boston, MA, (1994). · Zbl 0835.32001
[39] Joyce, D. Hypercomplex algebraic geometry,Quart. J. Math. Oxford Ser. (2),49(194), 129–162, (1998). · Zbl 0924.14002
[40] Joyce, D.Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs, Oxford University Press, Oxford, (2000). · Zbl 1027.53052
[41] Krylov, N.V. Nonlinear elliptic and parabolic equations of the second order, Translated from the Russian by P.L. Buzytskiî,Mathematics and its Applications, (Soviet Series),7, D. Reidel, Publishing Co., Dordrecht, (1987). · Zbl 0619.35004
[42] Krylov, N.V. Smoothness of the value function for a controlled diffusion process in a domain,Izvestiya Akademii Nauk SSSR, seriya matematicheskaya,53(1), 66–96, (1989), in Russian; English translation inMath. USSR Izvestija,34(1), (1990).
[43] Lelong, P. Fonctions plurisousharmoniques et formes différentielles positives. (French), Gordon & Breach, Paris-London-New York, (Distributed by Dunod éditeur, Paris), (1968). · Zbl 0195.11603
[44] Maxwell, J.C.The Scientific Letters and Papers of James Clerk Maxwell, Vol. II, 1862–1873, Harman, P.M., Ed., Cambridge University Press, (1995).
[45] Moisil, Gr.C. Sur les quaternions monogènes,Bull. Sci. Math.,55, 168–194, (1931). · JFM 57.0343.01
[46] Moore, E.H. On the determinant of an hermitian matrix of quaternionic elements,Bull. Am. Math. Soc.,28, 161–162, (1922). · JFM 48.0128.07
[47] Palamodov, V.P. Holomorphic synthesis of monogenic functions of several quaternionic variables,J. Anal. Math.,78, 177–204, (1999). · Zbl 0962.30027 · doi:10.1007/BF02791133
[48] Pertici, D. Quaternion regular functions and domains of regularityBoll. Un. Math. Ital. B (7),7(4), 973–988, (1993). · Zbl 0803.30040
[49] Pogorelov, A.V. The regularity of the generalized solutions of the equation det( u/ i j ) = (x 1,x 2,...x n ) > 0, (Russian),Dokl. Akad. Nauk SSSR,200, 534–537, (1971).
[50] Pogorelov, A.V. The Dirichlet problem for the multidimensional analogue of the Monge-Ampère equation, (Russian),Dokl. Akad. Nauk SSSR,201, 790–793, (1971). · Zbl 0238.35071
[51] Pogorelov, A.V. A regular solution of then-dimensional Minkowski problem,Dokl. Akad. Nauk SSSR,199, 785–788 (Russian); Translated asSoviet Math. Dokl.,12, 1192–1196, (1971).
[52] Pogorelov, A.V. Mnogomernoe uravnenie Monzha-Ampera det, ||z ij ||=(z 1,...,z n ,z, x 1,...,x n ), (Russian), The multidimensional Monge-Ampère equation det ||z ij ||=(z 1,...,z n ,z, x 1,...,x n ),Nauka, Moscow, (1988). · Zbl 0643.35004
[53] Quillen, D. Quaternionic algebra and sheaves on the Riemann sphere,Quart. J. Math. Oxford Ser. (2),49(194), 163–198, (1998). · Zbl 0921.14003
[54] Rachev, S.T. and Rüschendorf, L.Mass Transportation Problems, Probability and its Applications, Springer-Verlag, New York, (1998). · Zbl 0990.60500
[55] Rauch, J. and Taylor, B.A. The Dirichlet problem for the multidimensional Monge-Ampère equation,Rocky Mountain J. Math.,7(2), 345–364, (1977). · Zbl 0367.35025 · doi:10.1216/RMJ-1977-7-2-345
[56] Ronkin, L.I. Introduction to the theory of entire functions of several variables, Translated from the Russian by Israel Program for Scientific Translation, Translations of Mathematical Monographs, Vol. 44, American Mathematical Society, Providence, RI, (1974). · Zbl 0286.32004
[57] Rosenberg, J. AlgebraicK-theory and its applications,Graduate Texts in Mathematics,147, Springer-Verlag, New York, (1994).
[58] Sudbery, A. Quaternionic analysis,Math. Proc. Cambridge Philos. Soc.,85(2), 199–224, (1979). · Zbl 0399.30038 · doi:10.1017/S0305004100055638
[59] Vilenkin, N.Ja. Special functions and the theory of group representations, Translated from the Russian by V.N. Singh, Translations of Mathematical Monographs, Vol. 22, American Mathematical Society, Providence, RI, (1968). · Zbl 0172.18404
[60] Yau, S.-T. On Calabi’s conjecture and some new results in algebraic geometry,Proceedings of the National Academy of Sciences of the USA,74, 1798–1799, (1977). · Zbl 0355.32028 · doi:10.1073/pnas.74.5.1798
[61] Yau, S.-T. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I,Comm. Pure Appl. Math.,31(3), 339–411, (1978). · Zbl 0369.53059 · doi:10.1002/cpa.3160310304
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