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The complex Monge-Ampère equation with infinite boundary value. (English) Zbl 1139.32018

Authors’ abstract: In this article we consider the complex Monge-Ampère equation with infinite boundary value in bounded pseudoconvex domains. We prove the existence of strictly plurisubharmonic solution to the problem in convex domains under suitable growth conditions. We also obtain, for general pseudoconvex domains, some nonexistence results which show that these growth conditions are nearly optimal.

MSC:

32W20 Complex Monge-Ampère operators
32T15 Strongly pseudoconvex domains
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[1] Blocki, Z., Interior regularity of the complex Monge-Ampère equation in convex doamins, Duke Math. J., 105, 167-181 (2000) · Zbl 1020.32031
[2] Caffarelli, L.; Kohn, J. J.; Nirenberg, L.; Spruck, J., The Dirichlet problem for nonlinear second-order elliptic equations, II: Complex Monge-Ampère and uniformly elliptic equations, Comm. Pure Appl. Math., 209-252 (1985) · Zbl 0598.35048
[3] Cheng, S. Y.; Yau, S.-T., On the existence of a complete Kahler metric on noncompact complex manifolds and the regularity of Fefferman’s equation, Comm. Pure Appl. Math., 33, 507-544 (1980) · Zbl 0506.53031
[4] Cheng, S. Y.; Yau, S. T., The real Monge-Ampère equation and affine flat structure, (Chern, S. S.; Wu, W. T., Proc. 1980 Beijing Symp. on Diff. Geom. and Diff. Equations, vol. I (1982), Science Press: Science Press Beijing), 339-370
[5] Guan, B.; Jian, H.-Y., The Monge-Ampère equation with infinite boundary value, Pacific J. Math., 216, 77-94 (2004) · Zbl 1126.35318
[6] Ivarsson, B., Interior regularity of solutions to a complex Monge-Ampère equation, Ark. Mat., 40, 275-300 (2002) · Zbl 1066.32036
[7] Keller, J. B., On solutions of \(\triangle u = f(u)\), Comm. Pure Appl. Math., 10, 503-510 (1957) · Zbl 0090.31801
[8] Lions, P.-L., Sur les èquations de Monge-Ampère, I, Manuscripta Math., 41, 1-43 (1983) · Zbl 0509.35036
[9] Matero, J., The Bieberbach-Rademacher problem for the Monge-Ampère operator, Manuscripta Math., 91, 379-391 (1996) · Zbl 0873.35026
[10] Osserman, R., On the inequality \(\triangle u \geq f(u)\), Pacific J. Math., 7, 1641-1647 (1957) · Zbl 0083.09402
[11] Salani, P., Boundary blow-up problems for Hessian equations, Manuscripta Math., 96, 281-294 (1998) · Zbl 0907.35052
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