Curvature estimates for the level sets of solutions to the Monge-Ampère equation detD 2 u = 1

Chen, Chuanqiang; Ma, Xiang; Shi, Shujun

doi:10.1007/s11401-014-0864-6

Cited by 12 publications

(5 citation statements)

References 8 publications

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“…A maximum principle for ψ (defined in (3)), when κ = ν = n, has been proved by Feng and Shi (see [11], Theorem 1.1). When f = 1, the last result was proved in [4]. We discuss here the case of the P-function φ (defined in (2)).…”

Section: General ω ν = Nmentioning

confidence: 96%

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Maximum principles and overdetermined problems for Hessian equations

Enache¹,

Marras²,

Porru³

2021

APAM

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In this article we investigate some Hessian type equations. Our main aim is to derive new maximum principles for some suitable P-functions, in the sense of L.E. Payne, that is for some appropriate functional combinations of u(x) and its derivatives, where u(x) is a solution of the given Hessian type equations. To find the most suitable P-functions, we first investigate the special case of a ball, where the solution of our Hessian equations is radial, since this case gives good hints on the best functional to be considered later, for general domains. Next, we construct some elliptic inequalities for the well-chosen P-functions and make use of the classical maximum principles to get our new maximum principles. Finally, we consider some overdetermined problems and show that they have solutions when the underlying domain has a certain shape (spherical or ellipsoidal).

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Section: General ω ν = Nmentioning

confidence: 96%

“…Proof. Our method is very similar to that used in [4,11]. Since T (n) = O (the null matrix) and σ (n) = f , from the equation…”

Section: General ω ν = Nmentioning

confidence: 99%

Maximum principles and overdetermined problems for Hessian equations

Enache¹,

Marras²,

Porru³

2021

APAM

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“…∂ Ω  with 0-boundary Dirichlet condition are studied in reference [4]. At present, there are many methods to study the convexity of elliptic partial differential equation solutions and their level sets.…”

Section: U Umentioning

confidence: 99%

Curvature estimation of the level sets of solutions of a class of elliptic partial differential equation

2023

International Conference on Pure, Applied, and Computational Mathematics (PACM 2023)

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The curvature of the level set of elliptic partial differential equation solutions is always an important content in the study of convexity. Curvature is an important invariant of surface, which characterizes the degree of curve bending, is the important basis of differential geometry. Curvature is widely used in machining. In this paper, we study the completely nonlinear elliptic Monge-Ampère equation 2 det u D u e = with 0 boundary value Dirichlet condition in four-dimensional Euclidean space. It is proved that the auxiliary function obtains the maximum value at the boundary, and then the mean curvature and Gauss curvature of the level sets of the solutions of the equation are estimated quantitatively.

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“…If the solution u of the equation is strictly convex, it is a completely nonlinear and it is an elliptical partial differential equation. The equation 2 det 1 D u  is studied in reference [1] , they construct the auxiliary functions : For the class of completely nonlinear elliptic Monge-Ampère equations 2 det u D u e  , u D 2 is the Hessian matrix of u , the solution u is strictly convex when u D 2 is positive definite [2] . We will study the elliptic Monge-Ampère equations…”

Section: Introductionmentioning

confidence: 99%

Convexity Estimates for the Solution of the Elliptical Partial Differential Equation det D ² u = e^u

2023

J. Phys.: Conf. Ser.

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The Monge-Ampère equation det D 2 u = eu is completely nonlinear and elliptic, the convexity estimates for the solution of the elliptical partial differential equation is very important. We establish a differential inequality by constructing an auxiliary function and give two differential estimates for the solution of the equation det D 2 u = eu with 0 boundary value condition.

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Curvature estimates for the level sets of solutions to the Monge-Ampère equation detD 2 u = 1

Cited by 12 publications

References 8 publications

Maximum principles and overdetermined problems for Hessian equations

Maximum principles and overdetermined problems for Hessian equations

Curvature estimation of the level sets of solutions of a class of elliptic partial differential equation

Convexity Estimates for the Solution of the Elliptical Partial Differential Equation det D ² u = e^u

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Curvature estimates for the level sets of solutions to the Monge-Ampère equation detD 2 u = 1

Cited by 12 publications

References 8 publications

Maximum principles and overdetermined problems for Hessian equations

Maximum principles and overdetermined problems for Hessian equations

Curvature estimation of the level sets of solutions of a class of elliptic partial differential equation

Convexity Estimates for the Solution of the Elliptical Partial Differential Equation det D 2 u = eu

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Convexity Estimates for the Solution of the Elliptical Partial Differential Equation det D ² u = e^u