A weighted eigenvalue problem of the biased infinity Laplacian
                  <sup>*</sup>

Liu, Fang; Yang, Xiaoping

doi:10.1088/1361-6544/abd85d

Cited by 4 publications

(5 citation statements)

References 59 publications

(113 reference statements)

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“…They obtained existence, uniqueness and stability results for the boundary-value problem. Liu and Yang [21] established the existence of the principal Dirichlet eigenvalue based on the comparison principle. They also established the Harnack inequality and the Lipschitz regularity of a nonnegative viscosity supersolution to the β-biased equation…”

Section: Introductionmentioning

confidence: 99%

Lipschitz Regularity of Viscosity Solutions to the Infinity Laplace Equation

Han,

Liu

2023

JAMP

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In this paper, we study the viscosity solutions of the Neumann problemLaplacian. The normalized infinity Laplacian was first studied by Peres, Shramm, Sheffield and Wilson from the point of randomized theory named tug-of-war, which has wide applications in optimal mass transportation, financial option price problems, digital image processing, physical engineering, etc. We give the Lipschitz regularity of the viscosity solutions of the Neumann problem. The method we adopt is to choose suitable auxiliary functions as barrier functions and combine the perturbation method and viscosity solutions theory.

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Section: Introductionmentioning

confidence: 99%

Lipschitz Regularity of Viscosity Solutions to the Infinity Laplace Equation

Han,

Liu

2023

JAMP

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“…One can see [36] for more uniqueness results of infinity Laplacian equations. We direct the reader to [27,28,29,30,31,33,36,37,39,42,44] and the references therein for the ∞-Laplacian operator.…”

Section: Introductionmentioning

confidence: 99%

Viscosity solutions to the infinity Laplacian equation with lower terms

Li¹,

Liu²

2023

ejde

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We establish the existence and uniqueness of viscosity solutions tothe Dirichlet problem $$\displaylines{ \Delta_\infty^h u=f(x,u), \quad \hbox{in } \Omega,\cr u=q, \quad\hbox{on }\partial\Omega,}$$ where $q\in C(\partial\Omega)$, $h>1$, $\Delta_\infty^h u=|Du|^{h-3}\Delta_\infty u$. The operator $\Delta_\infty u=\langle D^2uDu,Du \rangle$ is the infinity Laplacian which is strongly degenerate, quasilinear and it is associated with the absolutely minimizing Lipschitz extension. When the nonhomogeneous term $f(x,t)$ is non-decreasing in $t$, we prove the existence of the viscosity solution via Perron's method. We also establish a uniqueness result based on the perturbation analysis of the viscosity solutions. If the function $f(x,t)$ is nonpositive (nonnegative) and non-increasing in $t$, we also give the existence of viscosity solutions by an iteration technique under the condition that the domain has small diameter. Furthermore, we investigate the existence and uniqueness of viscosity solutions to the boundary-value problem with singularity $$\displaylines{ \Delta_\infty^h u=-b(x)g(u), \quad \hbox{in } \Omega, \cr u>0, \quad \hbox{in } \Omega, \cr u=0, \quad \hbox{on }\partial\Omega, }$$ when the domain satisfies some regular condition. We analyze asymptotic estimates for the viscosity solution near the boundary.

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“…when the right hand side ( ) , f x t is non-decreasing in t and has one sign. In addition, it is also necessary to prove the comparison principle during the studies of the Dirichlet eigenvalue problem related to the infinity Laplacian, see for example [28] [29] [30].…”

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Uniqueness of Viscosity Solutions to the Dirichlet Problem Involving Infinity Laplacian

Sun,

Liu

2023

APM

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In this paper, we study the Dirichlet boundary value problem involving the highly degenerate and h-homogeneous quasilinear operator associated with the infinity Laplacian, where the right hand side term is ( ) ( ) , , n F x t p C ∈ Ω × ×   and the boundary value is ( ) C ϕ ∈ ∂Ω . First, we establish the comparison principle by the double variables method based on the viscosity solutions theory for the general equation ( ) , , h u F x u Du ∞ ∆ = in Ω . We propose two different conditions for the right hand side ( ) , , F x u Du and get the comparison principle results under different conditions by making different perturbations. Then, we obtain the uniqueness of the viscosity solution to the Dirichlet boundary value problem by the comparison principle. Moreover, we establish the local Lipschitz continuity of the viscosity solution.

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A weighted eigenvalue problem of the biased infinity Laplacian ^*

Cited by 4 publications

References 59 publications

Lipschitz Regularity of Viscosity Solutions to the Infinity Laplace Equation

Lipschitz Regularity of Viscosity Solutions to the Infinity Laplace Equation

Viscosity solutions to the infinity Laplacian equation with lower terms

Uniqueness of Viscosity Solutions to the Dirichlet Problem Involving Infinity Laplacian

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A weighted eigenvalue problem of the biased infinity Laplacian *

Cited by 4 publications

References 59 publications

Lipschitz Regularity of Viscosity Solutions to the Infinity Laplace Equation

Lipschitz Regularity of Viscosity Solutions to the Infinity Laplace Equation

Viscosity solutions to the infinity Laplacian equation with lower terms

Uniqueness of Viscosity Solutions to the Dirichlet Problem Involving Infinity Laplacian

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A weighted eigenvalue problem of the biased infinity Laplacian ^*