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The Dirichlet problem for the degenerate Monge-Ampère equation. (English) Zbl 0611.35029

The authors study the problem of finding a convex function u in \(\Omega\) such that \[ (1)\quad \det (u_{ij})=0\quad in\quad \Omega;\quad (2)\quad u=\phi \text{ given on }\partial \Omega, \] where \(\Omega\) is a bounded convex domain in \(R^ n\) with smooth, strictly convex boundary \(\partial \Omega\) and \(u_ i=\partial u/\partial x_ i\), \(u_{ij}=\partial^ 2u/\partial x_ i\partial x_ j\) etc. The existence of a smooth solution in \({\bar \Omega}\) satisfying (2) of the corresponding elliptic problem \[ (1)'\quad \det (u_{ij})=\psi >0\quad in\quad \Omega, \] has been recently shown by N. V. Krylov [Math. USSR, Izv. 22, 67-97 (1984); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 47, No.1, 75-108 (1983; Zbl 0578.35024)] and the authors [Commun. Pure Appl. Math. 37, 369-402 (1984; Zbl 0598.35047)] in case \(\psi\) and \(\phi\) are sufficiently smooth. It is interesting to treat the degenerate problem (1), (2). The corresponding question for degenerate complex Monge-Ampère equation to find a plurisubharmonic function w in a bounded pseudoconvex domain \(\Omega\) in \({\mathbb{C}}^ n\) satisfying \[ (3)\quad \det (w_{z_ j\bar z_ k})=0\quad in\quad \Omega, \] and (2) is also interesting. The authors with J. J. Kohn [Commun. Pure Appl. Math. 38, 209-252 (1985; Zbl 0598.35048)] treated the equation \[ (3)'\quad \det (w_{z_ j\bar z_ k})=\psi >0\quad in\quad \Omega, \] and showed that there is a plurisubharmonic solution w belonging to \(C^{1,1}(\Omega)\), provided \(\psi\not\equiv 0\), \(\psi\) satisfies some other conditions, and \(\psi\) and \(\phi\) are sufficiently smooth.
In the unique solution of (1), (2) is given by \[ (4)\quad u(x)=\max \{v(x)| \quad v\in C({\bar \Omega}),\text{ v convex and \(v\leq \phi\) on }\partial \Omega\}, \] and several authors have studied the regularity of u. The authors prove an extension up to the boundary of the regularity in case \(\phi\) is sufficiently smooth.
Reviewer: S.D.Bajpai

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J70 Degenerate elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35D10 Regularity of generalized solutions of PDE (MSC2000)
32U05 Plurisubharmonic functions and generalizations
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