The Boundary Regularity for Normalized Infinity Laplace Equations

Yanhui Li; Xiaotao Huang

doi:doi:10.11648/j.sd.20210906.19

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The Boundary Regularity for Normalized Infinity Laplace Equations

Yanhui Li, Xiaotao Huang

Published in Science Discovery (Volume 9, Issue 6)

Received: 11 October 2021 Accepted: 9 November 2021 Published: 12 November 2021

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Abstract

Infinity Laplace equations, which derive from minimal Lipschitz extensions and absolutely minimal variational problems, have been widely applied in zero-sum tug-of-war game, optimal transport, shape deformation and so on. However, due to the quasi-linearity, extreme degeneration (non-degeneration only in the gradient direction) and non-divergence of the equations, it is difficult to define its classical or weak solutions. After introducing the idea of viscosity solutions, the theoretical research of infinite Laplace equations begin to develop. We study the boundary Hölder regularity of solutions for inhomogeneous normalized infinite Laplace equations on bounded domains. Main ideas are as follows: Firstly, we get bounded estimate of solutions through constructing barrier functions about super(sub)-solutions. Secondly, we use iterative method to approach the solutions of equations. Finally, we obtain regularity estimates near the boundary by calculating error between barrier functions and solutions of equations. This paper proves that the visvosity solutions of inhomogeneous normalized infinite Laplace equations are Hölder continious on Lipschitz boundary provided that the region boundary is Lipschitz continuous, the right inhomogeneous term is positive (negative) continuous and the boundary values are Hölder continuous. On the basis of our conclusion, the global Hölder regularity theory of the normalized infinite Laplace equations can be obtained combining with the internal regularity estimates. In addition, this method can be extended to the boundary estimates of the infinite fractional Laplace equations.

Published in	Science Discovery (Volume 9, Issue 6)
DOI	10.11648/j.sd.20210906.19
Page(s)	329-334
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2021. Published by Science Publishing Group

Keywords

Normalized Infinity Laplace Equations, Boundary Regularity, Barrier Functions, Iterative Method

References

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[12]	B. Mebrate, A. Mohammed. Harnack inequality and an asymptotic mean-value property for the Finsler infinity-Laplacian [J]. Advances in Calculus of Variations, 2019, 14: 0-0.
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Yanhui Li, Xiaotao Huang. (2021). The Boundary Regularity for Normalized Infinity Laplace Equations. Science Discovery, 9(6), 329-334. https://doi.org/10.11648/j.sd.20210906.19

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ACS Style

Yanhui Li; Xiaotao Huang. The Boundary Regularity for Normalized Infinity Laplace Equations. Sci. Discov. 2021, 9(6), 329-334. doi: 10.11648/j.sd.20210906.19

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AMA Style

Yanhui Li, Xiaotao Huang. The Boundary Regularity for Normalized Infinity Laplace Equations. Sci Discov. 2021;9(6):329-334. doi: 10.11648/j.sd.20210906.19

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@article{10.11648/j.sd.20210906.19,
  author = {Yanhui Li and Xiaotao Huang},
  title = {The Boundary Regularity for Normalized Infinity Laplace Equations},
  journal = {Science Discovery},
  volume = {9},
  number = {6},
  pages = {329-334},
  doi = {10.11648/j.sd.20210906.19},
  url = {https://doi.org/10.11648/j.sd.20210906.19},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sd.20210906.19},
  abstract = {Infinity Laplace equations, which derive from minimal Lipschitz extensions and absolutely minimal variational problems, have been widely applied in zero-sum tug-of-war game, optimal transport, shape deformation and so on. However, due to the quasi-linearity, extreme degeneration (non-degeneration only in the gradient direction) and non-divergence of the equations, it is difficult to define its classical or weak solutions. After introducing the idea of viscosity solutions, the theoretical research of infinite Laplace equations begin to develop. We study the boundary Hölder regularity of solutions for inhomogeneous normalized infinite Laplace equations on bounded domains. Main ideas are as follows: Firstly, we get bounded estimate of solutions through constructing barrier functions about super(sub)-solutions. Secondly, we use iterative method to approach the solutions of equations. Finally, we obtain regularity estimates near the boundary by calculating error between barrier functions and solutions of equations. This paper proves that the visvosity solutions of inhomogeneous normalized infinite Laplace equations are Hölder continious on Lipschitz boundary provided that the region boundary is Lipschitz continuous, the right inhomogeneous term is positive (negative) continuous and the boundary values are Hölder continuous. On the basis of our conclusion, the global Hölder regularity theory of the normalized infinite Laplace equations can be obtained combining with the internal regularity estimates. In addition, this method can be extended to the boundary estimates of the infinite fractional Laplace equations.},
 year = {2021}
}

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TY  - JOUR
T1  - The Boundary Regularity for Normalized Infinity Laplace Equations
AU  - Yanhui Li
AU  - Xiaotao Huang
Y1  - 2021/11/12
PY  - 2021
N1  - https://doi.org/10.11648/j.sd.20210906.19
DO  - 10.11648/j.sd.20210906.19
T2  - Science Discovery
JF  - Science Discovery
JO  - Science Discovery
SP  - 329
EP  - 334
PB  - Science Publishing Group
SN  - 2331-0650
UR  - https://doi.org/10.11648/j.sd.20210906.19
AB  - Infinity Laplace equations, which derive from minimal Lipschitz extensions and absolutely minimal variational problems, have been widely applied in zero-sum tug-of-war game, optimal transport, shape deformation and so on. However, due to the quasi-linearity, extreme degeneration (non-degeneration only in the gradient direction) and non-divergence of the equations, it is difficult to define its classical or weak solutions. After introducing the idea of viscosity solutions, the theoretical research of infinite Laplace equations begin to develop. We study the boundary Hölder regularity of solutions for inhomogeneous normalized infinite Laplace equations on bounded domains. Main ideas are as follows: Firstly, we get bounded estimate of solutions through constructing barrier functions about super(sub)-solutions. Secondly, we use iterative method to approach the solutions of equations. Finally, we obtain regularity estimates near the boundary by calculating error between barrier functions and solutions of equations. This paper proves that the visvosity solutions of inhomogeneous normalized infinite Laplace equations are Hölder continious on Lipschitz boundary provided that the region boundary is Lipschitz continuous, the right inhomogeneous term is positive (negative) continuous and the boundary values are Hölder continuous. On the basis of our conclusion, the global Hölder regularity theory of the normalized infinite Laplace equations can be obtained combining with the internal regularity estimates. In addition, this method can be extended to the boundary estimates of the infinite fractional Laplace equations.
VL  - 9
IS  - 6
ER  -

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Author Information

Yanhui Li

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, China
Xiaotao Huang

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, China

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APA Style

Yanhui Li, Xiaotao Huang. (2021). The Boundary Regularity for Normalized Infinity Laplace Equations. Science Discovery, 9(6), 329-334. https://doi.org/10.11648/j.sd.20210906.19

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ACS Style

Yanhui Li; Xiaotao Huang. The Boundary Regularity for Normalized Infinity Laplace Equations. Sci. Discov. 2021, 9(6), 329-334. doi: 10.11648/j.sd.20210906.19

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AMA Style

Yanhui Li, Xiaotao Huang. The Boundary Regularity for Normalized Infinity Laplace Equations. Sci Discov. 2021;9(6):329-334. doi: 10.11648/j.sd.20210906.19

Copy | Download

@article{10.11648/j.sd.20210906.19,
  author = {Yanhui Li and Xiaotao Huang},
  title = {The Boundary Regularity for Normalized Infinity Laplace Equations},
  journal = {Science Discovery},
  volume = {9},
  number = {6},
  pages = {329-334},
  doi = {10.11648/j.sd.20210906.19},
  url = {https://doi.org/10.11648/j.sd.20210906.19},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sd.20210906.19},
  abstract = {Infinity Laplace equations, which derive from minimal Lipschitz extensions and absolutely minimal variational problems, have been widely applied in zero-sum tug-of-war game, optimal transport, shape deformation and so on. However, due to the quasi-linearity, extreme degeneration (non-degeneration only in the gradient direction) and non-divergence of the equations, it is difficult to define its classical or weak solutions. After introducing the idea of viscosity solutions, the theoretical research of infinite Laplace equations begin to develop. We study the boundary Hölder regularity of solutions for inhomogeneous normalized infinite Laplace equations on bounded domains. Main ideas are as follows: Firstly, we get bounded estimate of solutions through constructing barrier functions about super(sub)-solutions. Secondly, we use iterative method to approach the solutions of equations. Finally, we obtain regularity estimates near the boundary by calculating error between barrier functions and solutions of equations. This paper proves that the visvosity solutions of inhomogeneous normalized infinite Laplace equations are Hölder continious on Lipschitz boundary provided that the region boundary is Lipschitz continuous, the right inhomogeneous term is positive (negative) continuous and the boundary values are Hölder continuous. On the basis of our conclusion, the global Hölder regularity theory of the normalized infinite Laplace equations can be obtained combining with the internal regularity estimates. In addition, this method can be extended to the boundary estimates of the infinite fractional Laplace equations.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - The Boundary Regularity for Normalized Infinity Laplace Equations
AU  - Yanhui Li
AU  - Xiaotao Huang
Y1  - 2021/11/12
PY  - 2021
N1  - https://doi.org/10.11648/j.sd.20210906.19
DO  - 10.11648/j.sd.20210906.19
T2  - Science Discovery
JF  - Science Discovery
JO  - Science Discovery
SP  - 329
EP  - 334
PB  - Science Publishing Group
SN  - 2331-0650
UR  - https://doi.org/10.11648/j.sd.20210906.19
AB  - Infinity Laplace equations, which derive from minimal Lipschitz extensions and absolutely minimal variational problems, have been widely applied in zero-sum tug-of-war game, optimal transport, shape deformation and so on. However, due to the quasi-linearity, extreme degeneration (non-degeneration only in the gradient direction) and non-divergence of the equations, it is difficult to define its classical or weak solutions. After introducing the idea of viscosity solutions, the theoretical research of infinite Laplace equations begin to develop. We study the boundary Hölder regularity of solutions for inhomogeneous normalized infinite Laplace equations on bounded domains. Main ideas are as follows: Firstly, we get bounded estimate of solutions through constructing barrier functions about super(sub)-solutions. Secondly, we use iterative method to approach the solutions of equations. Finally, we obtain regularity estimates near the boundary by calculating error between barrier functions and solutions of equations. This paper proves that the visvosity solutions of inhomogeneous normalized infinite Laplace equations are Hölder continious on Lipschitz boundary provided that the region boundary is Lipschitz continuous, the right inhomogeneous term is positive (negative) continuous and the boundary values are Hölder continuous. On the basis of our conclusion, the global Hölder regularity theory of the normalized infinite Laplace equations can be obtained combining with the internal regularity estimates. In addition, this method can be extended to the boundary estimates of the infinite fractional Laplace equations.
VL  - 9
IS  - 6
ER  -

Copy | Download