Global C2‐Estimates for Convex Solutions of Curvature Equations

By Pengfei Guan, Changyu Ren, Zhizhang Wang. 2013

145 citations recorded by the source corpus.

We establish a C2 a priori estimate for convex hypersurfaces whose principal curvatures κ=(κ1,…, κn) satisfy σk(κ(X))=f(X,ν(X)), the Weingarten curvature equation. We also obtain such an estimate for admissible 2‐convex hypersurfaces in the case k=2. Our estimates resolve a longstanding problem in geometric fully nonlinear elliptic equations.© 2015 Wiley Periodicals, Inc.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9a469e65aab693dcdc9bb239e4449b523ca4dcd5

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Citations in the corpus

A unified flow approach to smooth, even Lp-Minkowski problems (2016, unknown): “w away all indices iwhich are neither pnor q. This gives term Bi. The ﬁrst term in the big bracket goes into Ai. In Corollary 14 of the appendix we will present an adaption of the method developed in [34] to deal with the curvature derivative terms Ai,Bi,Ci,Di,Ei. There we prove that we obtain the f”
Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds (2014, background): “It seems an interesting open question whether it is still true for f = σ 1/k k , 3 ≤ k < n; see [23].”
Interior C2 regularity of convex solutions to prescribing scalar curvature equations (2017, unknown): “ϑ ∈ Sn−1, P˜(x,ϑ) = 2logρ(x) +α( |∇u|2 2 )+β(x ·∇u−u) +loglogmax{uϑϑ,2}, where ρ(x) = 1 − |x|2, and α, β are constants to be determined later. The pick of test function P is inspired by a recent work [6]. The maximum value of P˜(x,β) in B¯ 1×Sn−1 must be attained in an interior point of B 1 since ρ ”
On estimates for the Fu–Yau generalization of a Strominger system (2015, unknown): “The C(2) estimate also builds in an essential way on the methods of [7], but we also exploit some new inequalities due to Guan, Ren, and Wang [10] in their work on real Hessian equations with gradient terms on the right hand side.”
A second order estimate for general complex Hessian equations (2015, methodology): “The real counterparts of (1-2) for 1 k ≤ n, with ψ not depending on the gradient of u, have been studied extensively in the literature (see the survey paper [Wang 2009] and more recent related work [Guan 2014]), as these equations appear naturally and play very important roles in both classical and”