On estimates for fully nonlinear parabolic equations on Riemannian manifolds

Guan, Bo; Shi, Shujun; Sui, Zhenan

doi:10.2140/apde.2015.8.1145

Cited by 24 publications

(27 citation statements)

References 19 publications

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“…The following lemma is crucial to construct barrier functions and the idea is mainly from [7,10] (see [12] also). …”

Section: Boundary Estimates For Second Order Derivativesmentioning

confidence: 99%

“…Similarly one can defineŨ (x, t) and λ ′ (Ũ (x, t)). The proof of (6.27) is similar to that of the elliptic case using an idea of Trudinger [27] (see [9,12]), so we only provide a sketch here. Definẽ…”

Section: Boundary Estimates For Second Order Derivativesmentioning

confidence: 99%

“…A generalization of (1.9) was considered in [17]. Guan, Shi and Sui [12] extended the work of [18] using the idea of [7]. As we know, works about the elliptic Hessian equations usually provide useful techniques to deal with the parabolic version.…”

Section: Introductionmentioning

confidence: 99%

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The first initial–boundary value problem for Hessian equations of parabolic type on Riemannian manifolds

Bao

Dong

Jiao

2016

Nonlinear Analysis: Theory, Methods & Applications

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MSC:Keywords: Riemannian manifolds Fully nonlinear parabolic equations First initial-boundary value problem a priori estimates a b s t r a c tIn this paper, we are concerned with the first initial-boundary value problem for a class of fully nonlinear parabolic equations on Riemannian manifolds. As usual, the establishment of the a priori C 2 estimates is our main part. Based on these estimates, the existence of classical solutions is proved under conditions which are nearly optimal.

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“…The following lemma is crucial to construct barrier functions and the idea is mainly from [7,10] (see [12] also). …”

Section: Boundary Estimates For Second Order Derivativesmentioning

confidence: 99%

Section: Boundary Estimates For Second Order Derivativesmentioning

confidence: 99%

See 1 more Smart Citation

The first initial–boundary value problem for Hessian equations of parabolic type on Riemannian manifolds

Bao

Dong

Jiao

2016

Nonlinear Analysis: Theory, Methods & Applications

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“…The author wished to thank Jiaping Wang and Xiangwen Zhang for stimulating communications and especially for pointing out mistakes in our previous attempts to prove the gradient estimates, and to thank Heming Jiao, Shujun Shi and Zhenan Sui for fruitful discussions and their contributions to our joint papers [19,22] where part of results in [18] were extended to more general elliptic and parabolic equations.…”

Section: Introductionmentioning

confidence: 99%

The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds

Guan¹,

Jiao²

2015

DCDS-A

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We solve the Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds under essentially optimal structure conditions, especially with no restrictions to the curvature of the underlying manifold and the second fundamental form of its boundary. The main result (Theorem 1.1) includes a new (and optimal) result in the Euclidean case. We introduce some new ideas and methods in deriving a priori estimates, which can be used to treat other types of fully nonlinear elliptic and parabolic equations on real or complex manifolds.Mathematical Subject Classification (2010): 35J15, 58J05, 35B45. Keywords: Fully nonlinear elliptic equations on Riemannian manifolds, Dirichlet problem, a priori estimates, concavity, subsolutions.

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“…The method is to apply the maximum principle to the evolution equation of τ ij [s] together with an auxiliary function involving the anisotropic support function s and its gradient. This is the most technical part in the proof, where an observation on smooth symmetric functions due to Guan, Shi and Sui [11] will be used.…”

mentioning

confidence: 99%

A fully nonlinear locally constrained anisotropic curvature flow

Wei¹,

Xiong²

2021

Preprint

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Given a smooth positive function F ∈ C ∞ (S n ) such that the square of its positive 1-homogeneous extension on R n+1 \ {0} is uniformly convex, the Wulff shape WF is a smooth uniformly convex body in the Euclidean space R n+1 with F being the support function of the boundary ∂WF . In this paper, we introduce the fully nonlinear locally constrained anisotropic curvature flow, where E k denotes the normalized kth anisotropic mean curvature with respect to the Wulff shape WF , σF the anisotropic support function and νF the outward anisotropic unit normal of the evolving hypersurface. We show that starting from a smooth, closed and strictly convex hypersurface in R n+1 (n ≥ 2), the smooth solution of the flow exists for all positive time and converges smoothly and exponentially to a scaled Wulff shape. A nice feature of this flow is that it improves a certain isoperimetric ratio. Therefore by the smooth convergence of the above flow, we provide a new proof of a class of the Alexandrov-Fenchel inequalities for anisotropic mixed volumes of smooth convex domains in the Euclidean space. ContentsReferences 2010 Mathematics Subject Classification. 53C44, 53C21.

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On estimates for fully nonlinear parabolic equations on Riemannian manifolds

Cited by 24 publications

References 19 publications

The first initial–boundary value problem for Hessian equations of parabolic type on Riemannian manifolds

The first initial–boundary value problem for Hessian equations of parabolic type on Riemannian manifolds

The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds

A fully nonlinear locally constrained anisotropic curvature flow

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