Levi-equation in higher dimensions

Zbginiew Slodkowski; Giuseppe Tomassini

Displaying similar documents to “Levi-equation in higher dimensions”

Levi equation and evolution of subsets of $C^{2}$

Zbigniew Slodkowski, Giuseppe Tomassini (1996)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In this Note we state some results obtained studying the evolution of compact subsets of $C^{2}$ by Levi curvature. This notion appears to be the natural extension to Complex Analysis of the notion of evolution by mean curvature.

The Dirichlet problem for the degenerate Monge-Ampère equation.

Luis A. Caffarelli, Louis Nirenberg, Joel Spruck (1986)

Revista Matemática Iberoamericana

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Let Ω be a bounded convex domain in Rⁿ with smooth, strictly convex boundary ∂Ω, i.e. the principal curvatures of ∂Ω are all positive. We study the problem of finding a convex function u in Ω such that: det (u_ij) = 0 in Ω u = φ given on ∂Ω.

Envelope of Dirichlet problems on a domain in $C^{n}$

Eric Bedford (1985)

Annales Polonici Mathematici

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Hölder regularity for solutions to complex Monge-Ampère equations

Mohamad Charabati (2015)

Annales Polonici Mathematici

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We consider the Dirichlet problem for the complex Monge-Ampère equation in a bounded strongly hyperconvex Lipschitz domain in ℂⁿ. We first give a sharp estimate on the modulus of continuity of the solution when the boundary data is continuous and the right hand side has a continuous density. Then we consider the case when the boundary value function is $^{1, 1}$ and the right hand side has a density in $L^{p} (Ω)$ for some p > 1, and prove the Hölder continuity of the solution.

On the Dirichlet problem in the Cegrell classes

Rafał Czyż, Per Åhag (2004)

Annales Polonici Mathematici

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Let μ be a non-negative measure with finite mass given by $φ (d d^{c} ψ) ⁿ$ , where ψ is a bounded plurisubharmonic function with zero boundary values and $φ \in L^{q} ((d d^{c} ψ) ⁿ)$ , φ ≥ 0, 1 ≤ q ≤ ∞. The Dirichlet problem for the complex Monge-Ampère operator with the measure μ is studied.

Smooth regularity for solutions of the Levi Monge-Ampère equation

Francesca Lascialfari, Annamaria Montanari (2001)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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We present a smooth regularity result for strictly Levi convex solutions to the Levi Monge-Ampère equation. It is a fully nonlinear PDE which is degenerate elliptic. Hence elliptic techniques fail in this situation and we build a new theory in order to treat this new topic. Our technique is inspired to those introduced in [3] and [8] for the study of degenerate elliptic quasilinear PDE’s related to the Levi mean curvature equation. When the right hand side has the meaning of total curvature...

A note on the regularity of the degenerate complex Monge-Ampère equation

Szymon Pliś (2010)

Annales Polonici Mathematici

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We prove the almost $^{1, 1}$ regularity of the degenerate complex Monge-Ampère equation in a special case.

On the smoothness of viscosity solutions of the prescribed Levi-curvature equation

Giovanna Citti, Ermanno Lanconelli, Annamaria Montanari (1999)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In this paper a $C^{\infty}$ -regularity result for the strong viscosity solutions to the prescribed Levi-curvature equation is announced. As an application, starting from a result by Z. Slodkowski and G. Tomassini, the $C^{\infty}$ -solvability of the Dirichlet problem related to the same equation is showed.