Local estimates for some fully nonlinear elliptic equations

By Sophie Chen. 2005

88 citations recorded by the source corpus.

We present a method to derive local estimates for some classes of fully nonlinear elliptic equations. The advantage of our method is that we derive Hessian estimates directly from $C^0$ estimates. Also, the method is flexible and can be applied to a large class of equations.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/41bf2b4d0bd8d5d30d007cd111ae6fa355a7a637

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Yanyan Li
2. Neil S. Trudinger
3. Xu-jia Wang
4. Bo Guan
5. Luc Nguyen
6. Weimin Sheng
7. Aobing Li
8. H. Jiao

In Computational Theory and Mathematics

1. Neil S. Trudinger
2. Yanyan Li
3. Bo Guan
4. Xu-jia Wang
5. Luc Nguyen
6. Aobing Li
7. Qinian Jin
8. Jeff Viaclovsky

Citations in the corpus

• A compactness theorem for a fully nonlinear Yamabe problem under a lower Ricci curvature bound (2012, methodology): “For this step in the case μ+Γ ≤ 1, we make use of Bishop’s comparison theorem, a Riemannian version of Hawking’s singularity theorem in relativity, a Liouville theorem in [27] and local gradient estimates for solutions of (6) ([20, 13, 29, 45]).”
• Prescribing symmetric functions of the eigenvalues of the Ricci tensor (2004, background): “Such estimates were recently proved by S. Chen [ Che05 ]:”
• Local gradient estimates of solutions to some conformally invariant fully nonlinear equations (2006, background): “general manifolds was extended by Chen in [5] to (f, Γ) which satisfy (1), (2), (4), (5) and (9).”
• A priori Estimates and Existence for a Class of Fully Nonlinear Elliptic Equations in Conformal Geometry* (2006, methodology): “The proofs in [Ch] and in this paper are completely different.”
• SECOND-ORDER ESTIMATES AND REGULARITY FOR FULLY NONLINEAR ELLIPTIC EQUATIONS ON RIEMANNIAN MANIFOLDS (2012, background): “[20], [29], and in conformal geometry in which there is a huge literature; see for instance [6], [7], [8], [12], [30], [31], [32], [33], [34], [40], [41], [42], [48], [53], [54] and references therein.”
• The Yamabe problem for higher order curvatures (2005, background): “6.5. Finally we mention that S. Chen [ Cn ] has found a difierent proof of the interior gradient estimate for solutions to the elliptic equations in Section 3. See [Li, W3] for”
• Complete Conformal Metrics of Negative Ricci Curvature on Compact Manifolds with Boundary (2008, background): “Their result has been extended to more general symmetric functions; see for instance [5], [12], [20], [19] and [28].”
• Estimates and existence results for a fully nonlinear Yamabe problem on manifolds with boundary (2006, background): “When (M, g) is locally conformally flat and (f,Γ) = (σ 1/k k ,Γk) for 1 ≤ k ≤ n, Guan-Wang [18] 3 and Li-Li [26] independently proved the existence of solutions of (1.8).”
• Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds (2014, background): “20); see also [6] for a simplified proof.”