Weak solutions to the complex Hessian equation

Zbigniew Blocki[1]

  • [1] Jagiellonian University, Institute of Mathematics, Reymonta 4, 30-059 Kraków (Pologne)

Annales de l’institut Fourier (2005)

  • Volume: 55, Issue: 5, page 1735-1756
  • ISSN: 0373-0956

Abstract

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

How to cite

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Blocki, Zbigniew. "Weak solutions to the complex Hessian equation." Annales de l’institut Fourier 55.5 (2005): 1735-1756. <http://eudml.org/doc/116230>.

@article{Blocki2005,
abstract = {We investigate the class of functions associated with the complex Hessian equation $(dd^cu)^m\wedge \omega ^\{n-m\}=0$.},
affiliation = {Jagiellonian University, Institute of Mathematics, Reymonta 4, 30-059 Kraków (Pologne)},
author = {Blocki, Zbigniew},
journal = {Annales de l’institut Fourier},
keywords = {Complex Hessian equation; plurisubharmonic functions},
language = {eng},
number = {5},
pages = {1735-1756},
publisher = {Association des Annales de l'Institut Fourier},
title = {Weak solutions to the complex Hessian equation},
url = {http://eudml.org/doc/116230},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Blocki, Zbigniew
TI - Weak solutions to the complex Hessian equation
JO - Annales de l’institut Fourier
PY - 2005
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 5
SP - 1735
EP - 1756
AB - We investigate the class of functions associated with the complex Hessian equation $(dd^cu)^m\wedge \omega ^{n-m}=0$.
LA - eng
KW - Complex Hessian equation; plurisubharmonic functions
UR - http://eudml.org/doc/116230
ER -

References

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  1. E. Bedford, B.A. Taylor, The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math. 37 (1976), 1-44 Zbl0315.31007MR445006
  2. E. Bedford, B.A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), 1-41 Zbl0547.32012MR674165
  3. Z. Blocki, Estimates for the complex Monge-Ampère operator, Bull. Pol. Acad. Sci. 41 (1993), 151-157 Zbl0795.32003MR1414762
  4. Z. Blocki, On the definition of the Monge-Ampère operator in ℂ 2 , Math. Ann. 328 (2004), 415-423 Zbl1060.32018MR2036329
  5. Z. Blocki, The domain of definition of the complex Monge-Ampère operator Zbl1102.32018
  6. B. Bojarski, T. Iwaniec, Another approach to Liouville theorem, Math. Nachr. 107 (1982), 253-262 Zbl0527.30013MR695751
  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. 38 (1985), 209-252 Zbl0598.35048MR780073
  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. 155 (1985), 261-301 Zbl0654.35031MR806416
  9. U. Cegrell, The general definition of the complex Monge-Ampère operator, Ann. Inst. Fourier 54 (2004), 159-179 Zbl1065.32020MR2069125
  10. J.-P. Demailly, Mesures de Monge-Ampère et mesures plurisousharmoniques, Math. Z. 194 (1987), 519-564 Zbl0595.32006MR881709
  11. J.-P. Demailly, Potential theory in several complex variables, (1991) 
  12. J.-P. Demailly, Monge-Ampère operators, Lelong numbers and intersection theory Complex analysis and geometry, Univ. Ser. Math., Plenum, New York (1993), 115-193 Zbl0792.32006MR1211880
  13. J.-P. Demailly, Complex Analytic and Differential Geometry, (1997) 
  14. L. Garding, An inequality for hyperbolic polynomials, J. Math. Mech. 8 (1959), 957-965 Zbl0090.01603MR113978
  15. P. Guan, X.-N. Ma, The Christoffel-Minkowski problem I: Convexity of solutions of a Hessian equation, Invent. Math. 151 (2003), 553-577 Zbl1213.35213MR1961338
  16. N. Ivochkina, N.S. Trudinger, X.-J. Wang, The Dirichlet problem for degenerate Hessian equations, Comm. Partial Diff. Equations 29 (2004), 219-235 Zbl1140.35418MR2038151
  17. T. Iwaniec, G. Martin, Geometric function theory and non-linear analysis, Clarendon Press (2001) Zbl1045.30011MR1859913
  18. M. Klimek, Pluripotential Theory, (1991), Clarendon Press Zbl0742.31001MR1150978
  19. N.V. Krylov, On the general notion of fully nonlinear second-order elliptic equations, Trans. Amer. Math. Soc. 347 (1995), 857-895 Zbl0832.35042MR1284912
  20. S.-Y. Li, On the Dirichlet problems for symmetric function equations of the eigenvalues of the complex Hessian, Asian J. Math. 8 (2004), 87-106 Zbl1068.32024MR2128299
  21. N.S. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995), 151-164 Zbl0887.35061MR1368245
  22. N.S. Trudinger, X.-J. Wang, Hessian measures II, Ann. of Math. 150 (1999), 579-604 Zbl0947.35055MR1726702
  23. J.B. Walsh, Continuity of envelopes of plurisubharmonic functions, J. Math. Mech. 18 (1968), 143-148 Zbl0159.16002MR227465
  24. A. Zeriahi, Volume and capacity of sublevel sets of a Lelong class of plurisubharmonic functions, Indiana Univ. Math. J. 50 (2001), 671-703 Zbl1138.31302MR1857051

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