### Citations

318 |
The Dirichlet problem for nonlinear second-order elliptic equations
- Caffarelli, Kohn, et al.
- 1985
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Citation Context ...nd, in the particular case of the Dirichlet problem (1)–(2), this sign is negative. The general technique employed in the solvability of the Dirichlet problem (1)–(2) is the method of continuity (see =-=[1]-=- in this context). We need that for the same boundary values j on @O; there exists r 0 AC N ð %OÞ whose graph is a K0-hypersurface. For each t in 0ptp1; we wish to find a solution r t AC 2;a ð %OÞ of ... |

95 |
Boundedly inhomogeneous elliptic and parabolic equations, Izv
- Krylov
- 1982
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Citation Context ... and for Dirichlet boundary conditions one can see [4–7,10,13], without claiming that this list of articles is complete. The first main theorem of existence if due to Caffarelli et al. [1] and Krylov =-=[8]-=-. They proved that if DCR n is a strictly convex planar domain, there exists a unique graph over D of constant Gauss curvature K; for K sufficiently small depending on the boundary data. Later, Guan a... |

68 | Hypersurfaces of constant curvature in space forms
- Rosenberg
- 1993
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Citation Context ...fiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ M2K 2 q n 1 þ rp K 1 n : ð5Þ It is worthwhile to point out that Rosenberg proved an height estimate of Kgraphs over planar domains of R nþ1 and K40 =-=[12]-=-. Exactly, if S is a K-graph over DCR n and @S @D; then the maximum height h that S can rise above the plane containing @S satisfies the inequality hp 1 : ð6Þ A second result refers to the C1 norm o... |

64 |
Improper affine hyperspheres of convex type and a generalization of a theorem by K
- Calabi
- 1958
(Show Context)
Citation Context ...ding only on K and K; such that if r is a positive solution of (1)–(2) with j 1; the following inequality holds: sup O jrrjpCðK; KÞ: ð8Þ The proofs of Theorems 1 and 2 are inspired by ideas of Calabi =-=[2]-=- and Pogorelov [11] by considering the Riemannian metric induced by the second fundamental form. We compute the Laplacian of the modulus and supporting functions defined on the K-hypersurface. We then... |

17 | Boundary value problems for surfaces of constant Gauss curvature - Hoffman, Rosenberg, et al. - 1992 |

17 |
The Neumann problem for equations of Monge-Ampère type
- Lions, Trudinger, et al.
- 1986
(Show Context)
Citation Context ...d solution of (1)–(2). In recent years, hypersurfaces of prescribed Gauss curvature have been subject to intensive studies. To mention a few examples, the Neumann boundary conditions is considered in =-=[9]-=-, and for Dirichlet boundary conditions one can see [4–7,10,13], without claiming that this list of articles is complete. The first main theorem of existence if due to Caffarelli et al. [1] and Krylov... |

17 | On locally locally convex hypersurfaces with boundary - Trudinger, Wang |

14 |
Boundary-value problems on S n for surfaces of constant Gauss curvature
- Guan, J
- 1993
(Show Context)
Citation Context ...auss–Kronecker curvature (briefly Gauss curvature), which can be represented as S fxðqÞ rðqÞq; qA %Og; where x denotes the position vector of S in R nþ1 ; and boundary values xðqÞ jðqÞq on @O: See =-=[4]-=- for details. Moreover, the orientation N on S is given by rrðqÞ rðqÞq NðxðqÞÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2ðqÞþjrrj 2 q :... |

13 | The existence of hypersurfaces of constant Gauss curvature with prescribed boundary - Guan, Spruck |

6 | Locally Convex Hypersurfaces of Prescribed Curvature and - Ivochkina, Tomi - 1998 |

6 | Hypersurfaces of Prescribed Gauss Curvature and Boundary in Riemannian Manifolds - Nehring - 1998 |

1 | The Gauss map and second fundamental form of surfaces
- lvez, Martínez
- 2000
(Show Context)
Citation Context ...sume that the Gauss curvature K is constant. With the above notation, the following identities hold: X n i1 /ririN; ejSðpÞ 0; for all 1pjpn; ð9Þ X n i1 /ririN; NSðpÞ nHðpÞ: ð10Þ Proof. (See also =-=[3]-=-). Denote by gij the metric of S; that is, gij /ei; ejS; g detðgijÞ and ðg ij Þ the inverse of ðgijÞ: Then riN Xn g ik ek: ð11Þ k1 Because feig is an orthonormal frame for the metric s; K 1 g... |

1 |
On a regular solution of the n-dimensional analogue of the Monge–Ampe
- Pogorelov
- 1971
(Show Context)
Citation Context ... K; such that if r is a positive solution of (1)–(2) with j 1; the following inequality holds: sup O jrrjpCðK; KÞ: ð8Þ The proofs of Theorems 1 and 2 are inspired by ideas of Calabi [2] and Pogorelov =-=[11]-=- by considering the Riemannian metric induced by the second fundamental form. We compute the Laplacian of the modulus and supporting functions defined on the K-hypersurface. We then apply the same tec... |