Curvature estimates for the level sets of solutions to a class of Monge–Ampère equations

Feng, Haiyu; Shi, Shujun

doi:10.1016/j.na.2018.08.007

Cited by 3 publications

(2 citation statements)

References 15 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].…”

Section: General ω ν = Nmentioning

confidence: 99%

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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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“…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].…”

Section: General ω ν = Nmentioning

confidence: 99%

“…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

View full text Add to dashboard Cite

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A Note on Monge–Ampère Equation in $${\mathbb {R}}^{2}$$

Enache

Porru

2021

Results Math

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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 a class of Monge–Ampère equations

Cited by 3 publications

References 15 publications

Maximum principles and overdetermined problems for Hessian equations

Maximum principles and overdetermined problems for Hessian equations

A Note on Monge–Ampère Equation in $${\mathbb {R}}^{2}$$

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 a class of Monge–Ampère equations

Cited by 3 publications

References 15 publications

Maximum principles and overdetermined problems for Hessian equations

Maximum principles and overdetermined problems for Hessian equations

A Note on Monge–Ampère Equation in $${\mathbb {R}}^{2}$$

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

Contact Info

Product

Resources

About

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