Journal ArticleDOI
C2 a priori estimates for degenerate Monge-Ampère equations
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This article is published in Duke Mathematical Journal.The article was published on 1997-01-01. It has received 61 citations till now. The article focuses on the topics: Ampere.read more
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A microscopic convexity principle for nonlinear partial differential equations
Baojun Bian,Pengfei Guan +1 more
TL;DR: In this paper, the authors studied the microscopic convexity property of fully nonlinear elliptic and parabolic partial differential equations and established that the rank of Hessian ∇cffff 2 istg u is of constant rank for any convex solution u of equation F(∇¯¯¯¯ 2¯¯ u,∇ u, u,u,x)=0.
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A gradient estimate in the Calabi–Yau theorem
TL;DR: In this paper, a C 1 -estimate for the complex Monge-Ampere equation on a compact Kahler manifold was obtained without using a C 2 -estimation.
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The Dirichlet Problem for degenerate Hessian Equations
TL;DR: In this paper, the Dirichlet problem for a class of fully nonlinear degenerate elliptic equations which depend only on the eigenvalues of the Hessian matrix was studied.
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On the dirichlet problem for degenerate Monge-Ampère equations
TL;DR: The authors proved global second derivative estimates for the Dirichlet problem for degenerate Monge-Ampere equations, which yield corresponding existence and regularity results, while drawing on previous investigations.
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The Existence of Hypersurfaces of Constant Gauss Curvature with Prescribed Boundary
Bo Guan,Joel Spruck +1 more
TL;DR: In this article, the problem of finding hypersurfaces of constant Gauss curvature (K-hypersurfaces) with prescribed boundary Γ in Rn+1, using the theory of Monge-Ampere equations was studied.
References
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The dirichlet problem for nonlinear second‐order elliptic equations I. Monge‐ampégre equation
TL;DR: On considere le probleme de Dirichlet as discussed by the authors for des equations elliptiques non lineaires for a fonction reelle u definie dans la fermeture d'un domaine borne Ω dans R n avec une frontiere ∂Ω C ∞
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Boundedly nonhomogeneous elliptic and parabolic equations in a domain
TL;DR: In this article, the first boundary value problem for equations of the form, where and are positive homogeneous functions of the first degree in, convex upwards in, that satisfy a uniform strict ellipticity condition is proved.
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The dirichlet problem for nonlinear second‐order elliptic equations. II. Complex monge‐ampère, and uniformaly elliptic, equations
TL;DR: On considere le probleme de Dirichlet pour des equations elliptiques non lineaires d'ordre 2 pour une fonction reelle dans un domaine borne Ω de R n a frontiere lisse ∂ Ω as discussed by the authors.
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Some counterexamples to the regularity of Monge-Ampère equations
TL;DR: In this article, it was shown that the solution u of the MongeAmpere equation det(D2u) = f(x), with u = 0 on the boundary, may not lie in W2,p or in C1, t for noncontinuous and positive f(X) and for continuous and nonnegative f(Y).