A Bernstein problem for special Lagrangian equations in exterior domains

Li, Dongsheng; Li, Zhisu; Yuan, Yu

doi:10.48550/arxiv.1709.04727

Cited by 2 publications

(11 citation statements)

References 15 publications

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“…In this section, we give an rigorous proof of Theorems 1.2, 1.3 and 1.4 based on Theorem 2.1 in [26]. For reading simplicity, we repeat the statement of this theorem.…”

Section: Constant Right Hand Side Situationmentioning

confidence: 98%

“…First, we introduce the following result on asymptotic behavior of classical solutions of special Lagrangian equation (1.1) with τ = π 2 . See Theorem 1.1 of [26]. Theorem 2.2.…”

Section: π 4 < τ < π 2 Situationmentioning

confidence: 99%

“…Now that we have obtained the limit of Hessian at infinity and it converge with a Hölder decay speed, it is time to capture the linear and constant part of the solution. In order to do this, we follow the line of Li-Li-Yuan [26] and analyze the linearized equation of (1.9). By the characterise result of ellipticity and concavity structure of F (λ(D 2 u)) type equation in [6], equation (1.9) is uniformly elliptic and concave.…”

Section: Analysis Of Linearized Equationmentioning

confidence: 99%

“…We prove the following results that provide an asymptotic behavior at infinity. The idea is to use Legendre transform and apply the strategies in [26].…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic behavior at infinity of solutions of Lagrangian mean curvature equations

Bao¹,

Liu²

2020

Preprint

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We studied the asymptotic behavior of solutions with quadratic growth condition of a class of Lagrangian mean curvature equations F τ (λ(D 2 u)) = f (x) in exterior domain, where f satisfies a given asymptotic behavior at infinity. When f (x) is a constant near infinity, it is not necessary to demand the quadratic growth condition anymore. These results are a kind of exterior Liouville theorem, and can also be regarded as an extension of theorems of Pogorelov [33], Flanders [16] and Yuan [42,43].

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“…In this section, we give an rigorous proof of Theorems 1.2, 1.3 and 1.4 based on Theorem 2.1 in [26]. For reading simplicity, we repeat the statement of this theorem.…”

Section: Constant Right Hand Side Situationmentioning

confidence: 98%

“…First, we introduce the following result on asymptotic behavior of classical solutions of special Lagrangian equation (1.1) with τ = π 2 . See Theorem 1.1 of [26]. Theorem 2.2.…”

Section: π 4 < τ < π 2 Situationmentioning

confidence: 99%

Section: Analysis Of Linearized Equationmentioning

confidence: 99%

“…We prove the following results that provide an asymptotic behavior at infinity. The idea is to use Legendre transform and apply the strategies in [26].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Asymptotic behavior at infinity of solutions of Lagrangian mean curvature equations

Bao¹,

Liu²

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…The Dirichlet problem for bounded domain was studied by Bartnik-Simon [BS82] and the isolated singularity problem was studied by Ecker [Ec86]. The exterior problem is a "complimentary" one for elliptic equations; see for example [Be51] [Si87] for minimal hypersurfaces, [CL03] for Monge-Ampere equation, [LLY17] for special Lagrangian equation and [HZ18] for infinity harmonic functions, besides the classic works such as [GS56] for linear ones. We study the exterior problem for the maximal surface equation in this paper.…”

Section: Introductionmentioning

confidence: 99%