The Dirichlet Problem for Degenerate Hessian Equations

Ivochkina, N. M.; Trudinger, Neil S.; Wang, Xu‐Jia

doi:10.1081/pde-120028851

Cited by 82 publications

(62 citation statements)

References 15 publications

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“…Krylov's estimation covers equation (5.3) as it is equivalent to a concave equation (5.4) below and so can be expressed as a Bellman equation. For Hessian equations (such as (5.4)) the proof in [18] was simplified in [14].…”

Section: Theorem 52mentioning

confidence: 99%

Convex solutions to the mean curvature flow

2011

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In this paper we study the classification of ancient convex solutions to the mean curvature flow in Rn+1 . An open problem related to the classification of type II singularities is whether a convex translating solution is k-rotationally symmetric for some integer 2 ≤ k ≤ n, namely whether its level set is a sphere or cylinder S k−1 × R n−k . In this paper we give an affirmative answer for entire solutions in dimension 2. In high dimensions we prove that there exist nonrotationally symmetric, entire convex translating solutions, but the blow-down in space of any entire convex translating solution is k-rotationally symmetric. We also prove that the blow-down in space-time of an ancient convex solution which sweeps the whole space R n+1 is a shrinking sphere or cylinder.

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Section: Theorem 52mentioning

confidence: 99%

Convex solutions to the mean curvature flow

2011

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“…The power d−1 of g was also shown to be optimal by an example in [23]. A modification of such example shows that m − 1 is the lowest possible power of g when there exist second derivatives estimates for solutions of the mHessian equations (see also recent [11], where the authors did a very good survey of the literature on the Hessian equations). As pointed out in [7], the techniques there rest on the fact that the solution is convex in case m = d, which in general does not always hold true for m-Hessian equations.…”

Section: Introductionmentioning

confidence: 96%

“…This article is closely related to [11], [7] and [15]. A second-order partial differential equation is called Hessian equation if it is of the form…”

Section: Introductionmentioning

confidence: 99%

Hessian Equations with Elementary Symmetric Functions

Dong

2006

Communications in Partial Differential Equations

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“…[8], [15], [16], [19], [21], [22]. It is clear that if u(z) = u(x + iy) is independent of y then it is m-subharmonic if and only if it is m-convex (see [22]).…”

Section: Weak Solutions To the Complex Hessian Equationmentioning

confidence: 99%