Interior Hessian estimates for a class of Hessian type equations

Chen, Chuanqiang; Dong, Weisong; Han, Fei

doi:10.1007/s00526-022-02385-3

Cited by 8 publications

(4 citation statements)

References 33 publications

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“…• For the k-Hessian equations, see [19,59,65,66,68,77,113,131,134]. • For the k-Hessian quotient equations, see [8,32,33,34,37,44,49,89,91,113] • For the Lagrangian mean curvature equation, see [10,11,19,36,90,127,137,139,143,147] In addition, there exist Pogorelov's type interior C 1,1 estimates for some equations. The smoothness requirements of the boundary can be relaxed.…”

Section: Prescribed Mean Curvature Equationmentioning

confidence: 99%

Boundary pointwise C1, and C2, regularity for fully nonlinear elliptic equations

Lian

Zhang

2020

Journal of Differential Equations

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Section: Prescribed Mean Curvature Equationmentioning

confidence: 99%

Boundary pointwise C1, and C2, regularity for fully nonlinear elliptic equations

Lian

Zhang

2020

Journal of Differential Equations

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“…dt and the spatial derivatives also coincide. Replacing θ by θ = log W − log(u − a) + Aρ, we see that θ attains a maximal value at (x 0 , t 0 ), where W satisfies the same differential equation in this point as h 1 1 . Thus, without loss of generality, we may pretend h 1 1 to be a scalar and then θ can be written as…”

Section: 2mentioning

confidence: 92%

“…is the principle curvature of the hypersurface. We list some important properties of σ k (λ), see [1,2] for more details. Firstly, we point out that σ k (λ) are really mixed Hessian operators.…”

Section: Now We Calculate the Evolution Of Hmentioning

confidence: 99%

“…the sign of the coefficient of the highest order term Q 2 is negative and the sign of the coefficient of the lower order term Q is positive. Then applying the maximum principle to (4.2), we can derive 1 C ≤ Q ≤ C, where C depends on f, α, β, δ and M 0 . By maximum principle again, if Q − γ has a sign at t = 0, then it has the sign for all time.…”

Section: Curvature Estiamtesmentioning

confidence: 99%

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A Class of inverse curvature type flows in Rn+1

Sheng,

Xue

2024

Preprint

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In this paper, we consider an expanding flow of smooth, closed, $(\eta,k)$-convex hypersurfaces in Euclidean $\mathbb{R}^{n+1}$ with speed $u^{\alpha}\rho^{\delta}\sigma_k^{-\frac{\beta}{k}}(\lambda(\eta))$, where $u, \rho$ are the support function and radical function of the hypersurface, respectively, $\alpha,\delta\in\mathbb{R}^1$, $\beta>0$, $k$ is an integer and $1 \leq k \leq n$, $\eta=Hg-h$, the first Newton transformation of the second fundamental form $h$, $\lambda(\eta)$ denote the eigenvalues of $g^{-1}\eta$. For $\alpha+\delta+\beta\leq 1$, we prove that the flow has a unique smooth and $(\eta,k)$-convex solution for all time, and converges smoothly after normalisation, to a sphere centered at the origin. Moreover, for $\alpha+\delta+\beta< 1$, we prove that the flow with the speed $fu^{\alpha}\rho^{\delta}\sigma_k^{-\frac{\beta}{k}}(\lambda(\eta))$ exists for all time, and converges smoothly after normalisation to a soliton, which is a solution of $fu^{\alpha-1}\rho^{\delta}\sigma_k^{-\frac{\beta}{k}}(\lambda(\eta))=\gamma$, provided that $f$ is a smooth positive function on $\mathbb{S}^n$ and $\gamma$ is a positive constant. What's more, we can use a more general flow to prove the existence of solution to a class of Hessian quotient equations again.

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Uniqueness of Solutions to a Class of Mixed Hessian Quotient Type Equations

Chen

2023

J Geom Anal

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Interior Hessian estimates for a class of Hessian type equations

Cited by 8 publications

References 33 publications

Boundary pointwise C1, and C2, regularity for fully nonlinear elliptic equations

Boundary pointwise C1, and C2, regularity for fully nonlinear elliptic equations

A Class of inverse curvature type flows in Rn+1

Uniqueness of Solutions to a Class of Mixed Hessian Quotient Type Equations

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