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

Enache, Cristian; Porru, Giovanni

doi:10.1007/s00025-020-01339-5

Cited by 2 publications

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References 10 publications

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“…We discuss here the case of the P-function φ (defined in (2)). When κ = ν = n = 2, a maximum principle has been obtained by the first and the third authors in [10]. In what follows we obtain an extension of their result to any higher dimension: Theorem 2.4.…”

Section: General ω ν = Nsupporting

confidence: 52%

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 = Nsupporting

confidence: 52%

Maximum principles and overdetermined problems for Hessian equations

Enache¹,

Marras²,

Porru³

2021

APAM

Self Cite

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“…Related to k-Hessian equations, if k = 1 the k-Hessian equations become the well-known Laplacian equations, and if k = N the k-Hessian equations become the Monge-Ampère equations. Concerning Laplacian equations and Monge-Ampère equations, there are a great number of research papers, see for examples [1,6,7,22] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Existence and nonexistence of entire k-convex radial solutions to Hessian type system

Ji-xian

2021

Adv Differ Equ

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

Cited by 2 publications

References 10 publications

Maximum principles and overdetermined problems for Hessian equations

Maximum principles and overdetermined problems for Hessian equations

Existence and nonexistence of entire k-convex radial solutions to Hessian type system

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