A Description of the Stability Cones Generated by Differential Operators of Monge-Ampère Type

Ivochkina, N. M.

doi:10.1070/sm1985v050n01abeh002828

Cited by 43 publications

(46 citation statements)

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“…From the point of view of this definition the research carried out in [13] appears somewhat mysterious and looks like the author found a right class of functions by chance. (An unsuccessful attempt to apply the above definition in the case of equation (1.3) is given in Example 2.21 below).…”

Section: Preliminariesmentioning

confidence: 99%

“…In a way, this cuts us off from the linear theory and raises the obscure problem of finding a "model" nonlinear function F0 for any particular F. For professionals in the field this problem is not too hard, and many authors prefer to use model equations while treating concrete equations (see, for instance, [2], [3], [13]), but for a "ready-to-use" theory this "cut off" is highly undesirable since applications may advance equations different from those which have already been investigated. However, in the above system of notions we cannot avoid this difficulty unless we can either understand how to make the continuation with respect to the parameter t in the situation when the set 9?…”

Section: Preliminariesmentioning

confidence: 99%

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On the General Notion of Fully Nonlinear Second-Order Elliptic Equations

Krylov¹

1995

Transactions of the American Mathematical Society

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Abstract. The general notion of fully nonlinear second-order elliptic equation is given. Its relation to so-called Bellman equations is investigated. A general existence theorem for the equations like Pm(uxixj) = JJ™^1 ck(x)Pk(uxixj) *s obtained as an example of an application of the general notion of fully nonlinear elliptic equations.We will be dealing with the following question: Given an equation It may look strange that we address such a question. Indeed, there are even books [9], [17] and many articles about the general theory of fully nonlinear elliptic equations. Therefore, very general results are available in the theory. However, on the other hand, it turns out that if an unexperienced reader meets a fully nonlinear second-order partial differential equation in his investigations and tries to get any information about its solvability from the literature, then almost certainly he fails to find what he needs, unless he considers an equation that is exactly one which had already been treated. The point is that in the general theory we consider nonlinear equations only of a special type, say, such that for any x e D, £ e Rd, uij, w,, u e R S\{\2 < F(uij+^,z)-F(uij, z) < S~x\c]\2, where z = (w,, u, x) and S is a positive constant. Obviously, even for the simplest Monge-Ampere equation, when F = det(w,;) -f(x), conditions of this type are not satisfied (for instance, for d > 2 the function det(M,;) grows much faster than linearly). Nevertheless, the general theory applies to this and many other special equations, and the reason for this is that in an appropriate class of functions they can be reduced to those considered in the general theory. We apply some techniques to include into our theory concrete equations such as the Monge-Ampere equation (real or complex) or more general Weingarten equations. These techniques are different in different cases, and by answering our main question here we want to show, in particular, what should be done for

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Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

On the General Notion of Fully Nonlinear Second-Order Elliptic Equations

Krylov¹

1995

Transactions of the American Mathematical Society

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“…The properties of the k-Hessian operator was well discussed in a numerous papers written as a first author by Ivochkina (see [7]- [10] and others). Moreover, this operator appear as an object of investigation by many remarkable geometers.…”

Section: Introductionmentioning

confidence: 99%

A necessary and a sufficient condition for the existence of the positive radial solutions to Hessian equations and systems with weights

Covei

2017

Acta Mathematica Scientia

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Abstract. In this article we consider the existence of positive radial solutions for Hessian equations and systems with weights and we give a necessary condition as well as a sufficient condition for a positive radial solution to be large. The method of proving theorems is essentially based on a successive approximation. Our results complete and improve a recently work published by Zhang and Zhou (Existence of entire positive k-convex radial solutions to Hessian equations and systems with weights, Applied Mathematics Letters, Volume 50, December 2015, Pages 48-55).

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“…These are well-known as ellipticity cones for the σ k equation, see [Gȧr59], [Ivo83], [CNS85]. We will say that a metric g is strictly k-admissible if the eigenvalues of…”

Section: A Fully Nonlinear Yamabe Problemmentioning

confidence: 99%