Abstract
Given an unbounded strongly pseudoconvex domain Ω and a continuous real valued function h defined on bΩ, we study the existence of a (maximal) plurisubharmonic function Φ on Ω such that Φ|b Ω = h.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bedford E., Taylor B.A. (1976). The Dirichlet problem for a complex Monge–Ampére equation. Invent. Math. 37(1): 1–44. MR 0445006 (56 #3351)
Bremermann H.J. (1959). On a generalized Dirichlet problem for plurisubharmonic functions and pseudo-convex domains. Characterization of Šilov boundaries. Trans. Am. Math. Soc. 91, 246–276. MR0136766 (25 #227)
Chirka E.M., Shcherbina N.V. (1995). Pseudoconvexity of rigid domains and foliations of hulls of graphs. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22(4): 707–735. MR1375316 (98f:32016)
De Bartolomeis P., Tomassini G. (1981) Traces of pluriharmonic functions. Comp. Math. 44, 29–39. (in English)
Dufresnoy, A.: Sur l’équation de Monge–Ampére complexe dans la boule de Cn. Ann. Inst. Fourier (Grenoble) 39(3), 773–775, (1989) (in French). MR1030849 (90k:32020)
Hunt L.R., Murray J.J. (1979) q-plurisubharmonic functions and a generalized Dirichlet problem. Mich. Math. J. 25(3): 299–316. (in English). MR 512901 (80b:32018)
Klimek, M.: Pluripotential Theory, vol. 6. The Clarendon Press Oxford University Press, New York (1991), no.3, xiv+266, (1991) (in English). MR1150978 (93h:32021)
Lupacciolu G. (1987) Valeurs au bord de fonctions holomorphes dans des domaines non bornés de Cn. C. R. Acad. Sci. Paris Sér. I Math. 304(3), 67–69
Gay R., Sebbar A. (1985) Division et extension dans l’algébre A ∞(Ω) d’un ouvert pseudo-convexe à bord lisse de Cn. Math. Z. 189(3): 421–447. (in French). MR 783566 (86e:32020)
Shcherbina N., Tomassini G. (1999). The Dirichlet problem for Levi-flat graphs over unbounded domains. Intern. Math. Res. Notices 3, 111–151. MR 1672246 (2000a:32018)
Slodkowski Z. (1984). The Bremermann–Dirichlet problem for q-plurisubharmonic functions. Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4) 11(2), 303–326. MR 764948 (86a:32038)
Tomassini G. (1983) Sur les algébres \(A^{0}(\bar D)\) et \(A^{\infty}(\bar D)\) d’un domaine pseudoconvexe non borné. Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4), 10(2): 243–256. (in French). MR 728435 (85c:32029)
Walsh, J.B.: Continuity of envelopes of plurisubharmonic functions. J. Math. Mech. 18, 143–148 (1968/1969). MR 0227465 (37 #3049)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the MURST project “Geometric Properties of Real and Complex Manifolds”.
Rights and permissions
About this article
Cite this article
Simioniuc, A., Tomassini, G. The Bremermann–Dirichlet problem for unbounded domains of \({\mathbb{C}}^n\) . manuscripta math. 126, 73–97 (2008). https://doi.org/10.1007/s00229-008-0167-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-008-0167-x