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 11 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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A class of Hessian quotient equations in warped product manifolds

Sheng,

Xue

2024

Sci. China Math.

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Interior Hessian estimates for a class of Hessian type equations

Cited by 11 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

A class of Hessian quotient equations in warped product manifolds

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