A priori estimate for convex solutions to special Lagrangian equations and its application

Chen, Jingyi; Yuan, Yu; Warren, Micah

doi:10.1002/cpa.20261

Cited by 35 publications

(22 citation statements)

References 21 publications

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“…All solutions to special Lagrangian equation (1.1) with critical phase and with quadratic growth near infinity must have the same quadratic asymptotic behavior, if one combines the a priori gradient and Hessian estimates in [WY09a, WY10,WdY14] with our general exterior Liouville Theorem 2.1. This exterior Liouville type result and the exterior Bernstein Theorem 1.1 also hold true for continuous viscosity solutions, in light of the regularity for solutions of special Lagrangian equations [WY09a,WY09b,WY10,CWY09,WdY14].…”

Section: Introductionmentioning

confidence: 71%

A Bernstein problem for special Lagrangian equations in exterior domains

Yuan

2020

Advances in Mathematics

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We establish quadratic asymptotics for solutions to special Lagrangian equations with supercritical phases in exterior domains. The method is based on an exterior Liouville type result for general fully nonlinear elliptic equations toward constant asymptotics of bounded Hessian, and also certain rotation arguments toward Hessian bound. Our unified approach also leads to quadratic asymptotics for convex solutions to Monge-Ampère equations (previously known), quadratic Hessian equations, and inverse harmonic Hessian equations over exterior domains.

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Section: Introductionmentioning

confidence: 71%

A Bernstein problem for special Lagrangian equations in exterior domains

Yuan

2020

Advances in Mathematics

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“…We would like to remark that, for the interior Dirichlet problems there have been much extensive studies, see for example [CIL92], [CNS85], [Ivo85], [Kry83], [Urb90], [Tru90] and [Tru95]; see [BCGJ03] and the references given there for more on the Hessian quotient equations; and for more on the special Lagrangian equations, we refer the reader to [HL82], [Fu98], [Yuan02], [CWY09] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

The exterior Dirichlet problem for Hessian quotient equations

Li¹,

Dai²

2012

Journal of Mathematical Analysis and Applications

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In this paper, we establish the existence and uniqueness theorem for solutions of the exterior Dirichlet problem for Hessian quotient equations with prescribed asymptotic behavior at infinity. This extends the previous related results on the Monge-Ampère equations and on the Hessian equations, and rearranges them in a systematic way. Based on the Perron's method, the main ingredient of this paper is to construct some appropriate subsolutions of the Hessian quotient equation, which is realized by introducing some new quantities about the elementary symmetric functions and using them to analyze the corresponding ordinary differential equation related to the generalized radially symmetric subsolutions of the original equation.

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“…Bao-Chen [2] got Hessian estimates in terms of certain integrals of the Hessian for solutions to (1.1) with n = 3, Θ = π. Warren-Yuan obtained Hessian estimates of (1.1) in terms of gradients for solutions to (1.1) in the following cases: i) the solutions satisfies (1.2) with small gradients in [29]; ii) n = 2 in [31]; iii) n = 3 and |Θ| ≥ π 2 in [30,32]. For general n, Chen-Warren-Yuan [8] derived a priori interior Hessian estimates for smooth convex solutions to (1.1) (see the very recent work [7] for convex viscosity solutions). In [27], Wang-Yuan obtained a priori interior Hessian estimates for all the solutions to (1.1) with critical and supercritical phases in dimensions ≥ 3.…”

Section: Let U Be a Smooth Function On An Open Set ωmentioning

confidence: 99%

Liouville type theorems and Hessian estimates for special Lagrangian equations

Ding¹

2019

Preprint

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In this paper, we get a Liouville type theorem for the special Lagrangian equation with a certain 'convexity' condition, where Warren-Yuan first studied the condition in [29]. Based on Warren-Yuan's work, our strategy is to show a global Hessian estimate of solutions via the Neumann-Poincaré inequality on special Lagrangian graphs, and mean value inequality for superharmonic functions on these graphs. Moreover, we derive interior Hessian estimates in terms of the linear exponential dependence on the gradient of the solutions to the equation with this 'convexity' condition or with supercritical phase. Here, the linear exponential dependence is optimal.

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A priori estimate for convex solutions to special Lagrangian equations and its application

Abstract: We derive a priori interior Hessian estimates for special Lagrangian equations when the potential is convex. When the phase is very large, we show that continuous viscosity solutions are smooth in the interior of the domain.

Cited by 35 publications

References 21 publications

A Bernstein problem for special Lagrangian equations in exterior domains

A Bernstein problem for special Lagrangian equations in exterior domains

The exterior Dirichlet problem for Hessian quotient equations

Liouville type theorems and Hessian estimates for special Lagrangian equations

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