Journal ArticleDOI
Solution of the Dirichlet Problem for Curvature Equations of Order m
TLDR
In this paper, the first-order curvature equation coincides with the Monge-Ampere equation, and the second-order equation with the first order curvature equations of order.Abstract:
Solvability conditions for curvature equations of order which are sufficient, and almost necessary, are obtained, and theorems concerning the existence of solutions in , , , are proved. The first-order curvature equation coincides with the curvature equation of order , and the curvature equation of order with the Monge-Ampere equation.Bibliography: 18 titles.read more
Citations
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Journal ArticleDOI
On the Dirichlet problem for Hessian equations
TL;DR: In this paper, Calfarelli, Nirenberg and Spruek showed the existence of classical solutions for the Dirichiet problem under various hypotheses on the function f and the domain ft.
Journal ArticleDOI
Second-order estimates and regularity for fully nonlinear elliptic equations on riemannian manifolds
TL;DR: In this paper, a priori second-order estimates for solutions of a class of fully nonlinear elliptic equations on Riemannian manifolds under structure conditions which are close to optimal were derived.
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On the general notion of fully nonlinear second-order elliptic equations
TL;DR: In this paper, the authors considered the problem of fully nonlinear elliptic second-order partial differential equations and showed that the general theory applies to this and many other special equations, such as the Weingarten equations.
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On some inequalities for elementary symmetric functions
Mi Lin,Neil S. Trudinger +1 more
TL;DR: In this paper, the authors prove certain inequalities for elementary symmetric funtions that are relevant to the study of partial differential equations associated with curvature problems, which is relevant to our work.
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The Dirichlet problem for Hessian equations on Riemannian manifolds
TL;DR: The Dirichlet problem on Riemannian manifolds has been studied in this paper, where the authors assume a smooth function defined in an open, convex, symmetric cone with vertex at the origin.
References
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Book
Elliptic Partial Differential Equations of Second Order
David Gilbarg,Neil S. Trudinger +1 more
TL;DR: In this article, Leray-Schauder and Harnack this article considered the Dirichlet Problem for Poisson's Equation and showed that it is a special case of Divergence Form Operators.
Journal ArticleDOI
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 ∞
Journal ArticleDOI
The Dirichlet problem for nonlinear second order elliptic equations. III: Functions of the eigenvalues of the Hessian
TL;DR: On etudie le probleme de Dirichlet dans un domaine borne Ω de R n a frontiere lisse ∂Ω:F(D 2 u)=ψ dans Ω, u=φ sur ∂ Ω as discussed by the authors.