A variational theory of the Hessian equation

By Kai-Seng Chou, Xu-jia Wang. 2001

314 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b7db2673448f379484e75ddd1c7cd332baabe546

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

The Lp-Minkowski problem and the Minkowski problem in centroaffine geometry (2006, unknown): “From our work [ CW3 ] on the Hessian equations, it is clear that p = ini1 is the criti-”
SECOND-ORDER ESTIMATES AND REGULARITY FOR FULLY NONLINEAR ELLIPTIC EQUATIONS ON RIEMANNIAN MANIFOLDS (2012, methodology): “The author also wishes to thank Jiaping Wang for helpful discussions on the proof of Theorem 2.4 and related topics.”
Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds (2014, methodology): “2) in R for was first studied by Ivochkina [30] and Caffarelli, Nirenberg and Spruck [4], followed by work in [37], [51], [12], [46], [48], and [9], etc.”
A priori estimate for convex solutions to special Lagrangian equations and its application (2009, methodology): “Hessian estimates for solutions with certain strict convexity constraints to MongeAmpère equations and k equations with k 2 were obtained by Pogorelov [13] and Chou and Wang [5], respectively.”
Gauduchon metrics with prescribed volume form (2015, methodology): “We remark that an estimate of this form was proved in the context of the complex Hessian equations by Hou– Ma–Wu [20], making use of ideas of Chou–Wang [5].”
A second order estimate for complex Hessian equations on a compact Kähler manifold (2010, methodology): “In the following lemma, part (1) was proved by Chou–Wang [5, p. 1037].”
The Yamabe problem for higher order curvatures (2005, methodology): “y v(x) = (1+ |x|2)(2k−n)/2k (2.20) when l= 0, k<n 2 , and Ω = R n; and by the unique solution of σ k σ l (λ(D2v)) = 1 in Ω (2.21) for 1 ≤ l<k≤ n. ¿From the inequalities (2.17), it was proved in [CW2] that the Dirichlet problem σ k(λ(D2v)) = |v|p + f(v) in Ω, (2.22) v= 0 on ∂Ω, admits a nonzero”
Liouville property and regularity of a Hessian quotient equation (2003, methodology): “We next apply the interior second derivative estimates due to Chou-Wang [10], combined with (3.”
Hessian equations on closed Hermitian manifolds (2015, background): “Sinceλ| j ∈ Γ2, by the proof in [2] which used the result in [7], there exists a constant θ1 such thatσ1 (λ | j ) ≥ θ1λ2 if j = 1 andσ1 (λ | j ) ≥ θ1λ1 if 2 ≤ j ≤ n.”
On the curvature estimates for Hessian equations (2016, background): “More reference about these type of estimates can be found in [13, 30].”
Hessian equations with Infinite Dirichlet Boundary Value (2006, methodology): “One helpful approach is to use the methods on gradient estimates in [5].”