Weak solutions to the complex Hessian equation
[Solutions faibles associée au Hessien complexe]
Blocki, Zbigniew
Annales de l'Institut Fourier, Tome 55 (2005), p. 1735-1756 / Harvested from Numdam

Nous cherchons la classe de fonctions associées à l’équation complexe (dd c u) m ω n-m =0.

We investigate the class of functions associated with the complex Hessian equation (dd c u) m ω n-m =0.

Publié le : 2005-01-01
DOI : https://doi.org/10.5802/aif.2137
Classification:  32U05,  35J60
Mots clés: Hessien complexe, fonctions plurisousharmoniques
@article{AIF_2005__55_5_1735_0,
     author = {Blocki, Zbigniew},
     title = {Weak solutions to the complex Hessian equation},
     journal = {Annales de l'Institut Fourier},
     volume = {55},
     year = {2005},
     pages = {1735-1756},
     doi = {10.5802/aif.2137},
     mrnumber = {2172278},
     zbl = {1081.32023},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2005__55_5_1735_0}
}
Blocki, Zbigniew. Weak solutions to the complex Hessian equation. Annales de l'Institut Fourier, Tome 55 (2005) pp. 1735-1756. doi : 10.5802/aif.2137. http://gdmltest.u-ga.fr/item/AIF_2005__55_5_1735_0/

[1] E. Bedford; B.A. Taylor The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math., Tome 37 (1976), pp. 1-44 | Article | MR 445006 | Zbl 0315.31007

[2] E. Bedford; B.A. Taylor A new capacity for plurisubharmonic functions, Acta Math., Tome 149 (1982), pp. 1-41 | Article | MR 674165 | Zbl 0547.32012

[3] Z. Blocki Estimates for the complex Monge-Ampère operator, Bull. Pol. Acad. Sci., Tome 41 (1993), pp. 151-157 | MR 1414762 | Zbl 0795.32003

[4] Z. Blocki On the definition of the Monge-Ampère operator in 2 , Math. Ann., Tome 328 (2004), pp. 415-423 | Article | MR 2036329 | Zbl 1060.32018

[5] Z. Blocki The domain of definition of the complex Monge-Ampère operator (preprint. Amer. J. Math. (to appear), http://www.im.uj.edu.pl/~blocki/publ/) | Zbl 1102.32018

[6] B. Bojarski; T. Iwaniec Another approach to Liouville theorem, Math. Nachr., Tome 107 (1982), pp. 253-262 | Article | MR 695751 | Zbl 0527.30013

[7] L. Caffarelli; J.J. Kohn; L. Nirenberg; J. Spruck The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge-Ampère, and uniformly elliptic, equations, Comm. Pure Appl. Math., Tome 38 (1985), pp. 209-252 | Article | MR 780073 | Zbl 0598.35048

[8] L. Caffarelli; L. Nirenberg; J. Spruck The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian, Acta Math., Tome 155 (1985), pp. 261-301 | Article | MR 806416 | Zbl 0654.35031

[9] U. Cegrell The general definition of the complex Monge-Ampère operator, Ann. Inst. Fourier, Tome 54 (2004), pp. 159-179 | Article | Numdam | MR 2069125 | Zbl 1065.32020

[10] J.-P. Demailly Mesures de Monge-Ampère et mesures plurisousharmoniques, Math. Z., Tome 194 (1987), pp. 519-564 | Article | MR 881709 | Zbl 0595.32006

[11] J.-P. Demailly Potential theory in several complex variables (1991) (preprint)

[12] J.-P. Demailly Monge-Ampère operators, Lelong numbers and intersection theory Complex analysis and geometry, Tome Univ. Ser. Math., Plenum, New York (1993), pp. 115-193 | MR 1211880 | Zbl 0792.32006

[13] J.-P. Demailly Complex Analytic and Differential Geometry (1997) (, http://www-fourier.ujf-grenoble.fr/~demailly/books.html)

[14] L. Garding An inequality for hyperbolic polynomials, J. Math. Mech., Tome 8 (1959), pp. 957-965 | MR 113978 | Zbl 0090.01603

[15] P. Guan; X.-N. Ma The Christoffel-Minkowski problem I: Convexity of solutions of a Hessian equation, Invent. Math., Tome 151 (2003), pp. 553-577 | Article | MR 1961338 | Zbl 01965472

[16] N. Ivochkina; N.S. Trudinger; X.-J. Wang The Dirichlet problem for degenerate Hessian equations, Comm. Partial Diff. Equations, Tome 29 (2004), pp. 219-235 | MR 2038151 | Zbl 02130233

[17] T. Iwaniec; G. Martin Geometric function theory and non-linear analysis, Clarendon Press (2001) | MR 1859913 | Zbl 1045.30011

[18] M. Klimek Pluripotential Theory, Clarendon Press (1991) | MR 1150978 | Zbl 0742.31001

[19] N.V. Krylov On the general notion of fully nonlinear second-order elliptic equations, Trans. Amer. Math. Soc., Tome 347 (1995), pp. 857-895 | Article | MR 1284912 | Zbl 0832.35042

[20] S.-Y. Li On the Dirichlet problems for symmetric function equations of the eigenvalues of the complex Hessian, Asian J. Math., Tome 8 (2004), pp. 87-106 | MR 2128299 | Zbl 02148243

[21] N.S. Trudinger On the Dirichlet problem for Hessian equations, Acta Math., Tome 175 (1995), pp. 151-164 | Article | MR 1368245 | Zbl 0887.35061

[22] N.S. Trudinger; X.-J. Wang Hessian measures II, Ann. of Math., Tome 150 (1999), pp. 579-604 | Article | MR 1726702 | Zbl 0947.35055

[23] J.B. Walsh Continuity of envelopes of plurisubharmonic functions, J. Math. Mech., Tome 18 (1968), pp. 143-148 | MR 227465 | Zbl 0159.16002

[24] A. Zeriahi Volume and capacity of sublevel sets of a Lelong class of plurisubharmonic functions, Indiana Univ. Math. J., Tome 50 (2001), pp. 671-703 | MR 1857051 | Zbl 01679413