scholarly journals Comparison principle to the infinity Laplacian equation with lower term

2021 ◽  
Author(s):  
Cuicui Li ◽  
Fang Liu
Keyword(s):  
Comparison Principle ◽  
Lower Term ◽  
Infinity Laplacian ◽  
Laplacian Equation

Parabolic Biased Infinity Laplacian Equation Related to the Biased Tug-of-War

2019 ◽  
Vol 19 (1) ◽  
pp. 89-112 ◽  
Author(s):  
Fang Liu ◽  
Feida Jiang
Keyword(s):  
Viscosity Solutions ◽  
Comparison Principle ◽  
Laplacian Operator ◽  
Inhomogeneous Term ◽  
Infinity Laplacian ◽  
Lipschitz Continuous ◽  
Laplacian Equation ◽  
The Game Theory ◽  
Fixed Constant ◽  
Dirichlet Boundary Problem

Abstract In this paper, we study the parabolic inhomogeneous β-biased infinity Laplacian equation arising from the β-biased tug-of-war {u_{t}}-\Delta_{\infty}^{\beta}u=f(x,t), where β is a fixed constant and {\Delta_{\infty}^{\beta}} is the β-biased infinity Laplacian operator \Delta_{\infty}^{\beta}u=\Delta_{\infty}^{N}u+\beta\lvert Du\rvert related to the game theory named β-biased tug-of-war. We first establish a comparison principle of viscosity solutions when the inhomogeneous term f does not change its sign. Based on the comparison principle, the uniqueness of viscosity solutions of the Cauchy–Dirichlet boundary problem and some stability results are obtained. Then the existence of viscosity solutions of the corresponding Cauchy–Dirichlet problem is established by a regularized approximation method when the inhomogeneous term is constant. We also obtain an interior gradient estimate of the viscosity solutions by Bernstein’s method. This means that when f is Lipschitz continuous, a viscosity solution u is also Lipschitz in both the time variable t and the space variable x. Finally, when {f=0} , we show some explicit solutions.

scholarly journals Weak Comparison Principle for Weighted Fractional p -Laplacian Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Jin Xie
Keyword(s):  
Comparison Principle ◽  
Nonlinear Term ◽  
Laplacian Equation ◽  
Carathéodory Function ◽  
Weak Comparison Principle

The aim of this paper is to establish a weak comparison principle for a class fractional p -Laplacian equation with weight. The nonlinear term f x , s > 0 is a Carathéodory function which is possibly unbounded both at the origin and at infinity and such that f x , s s 1 − p decreases with respect to s for a.e. x ∈ Ω .

scholarly journals Infinity Laplacian equation with strong absorptions

2016 ◽  
Vol 270 (6) ◽  
pp. 2249-2267 ◽  
Author(s):  
Damião J. Araújo ◽  
Raimundo Leitão ◽  
Eduardo V. Teixeira
Keyword(s):  
Infinity Laplacian ◽  
Laplacian Equation

ZERO ORDER PERTURBATIONS TO FULLY NONLINEAR EQUATIONS: COMPARISON, EXISTENCE AND UNIQUENESS

2009 ◽  
Vol 11 (01) ◽  
pp. 131-164 ◽  
Author(s):  
FERNANDO CHARRO ◽  
IRENEO PERAL
Keyword(s):  
Nonlinear Equations ◽  
Comparison Principle ◽  
Existence And Uniqueness ◽  
Existence Of Solutions ◽  
Elliptic Operators ◽  
Zero Order ◽  
Infinity Laplacian ◽  
Fully Nonlinear Equations ◽  
Fully Nonlinear ◽  
First Case

We study existence of solutions to [Formula: see text] where F is elliptic and homogeneous of degree m, and either f(λ,u) = λ uqor f(λ,u) = λ uq+ ur, for 0 < q < m < r, and λ > 0. Furthermore, in the first case, we obtain that the solution is unique as a consequence of a comparison principle up to the boundary. Several examples, including uniformly elliptic operators and the infinity-laplacian, are considered.

Nonlocal infinity Laplacian equation on graphs with applications in image processing and machine learning

2014 ◽  
Vol 102 ◽  
pp. 153-163 ◽  
Author(s):  
Elmoataz Abderrahim ◽  
Desquesnes Xavier ◽  
Lakhdari Zakaria ◽  
Lézoray Olivier
Keyword(s):  
Machine Learning ◽  
Image Processing ◽  
Infinity Laplacian ◽  
Laplacian Equation

scholarly journals $L^{\infty }$ decay estimates of solutions of nonlinear parabolic equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hui Wang ◽  
Caisheng Chen
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Nonlinear Parabolic Equation ◽  
Divergence Form ◽  
Decay Estimates ◽  
Laplacian Equation ◽  
Estimates Of Solutions ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear ◽  
Doubly Nonlinear Parabolic Equation

AbstractIn this paper, we are interested in $L^{\infty }$ L ∞ decay estimates of weak solutions for the doubly nonlinear parabolic equation and the degenerate evolution m-Laplacian equation not in the divergence form. By a modified Moser’s technique we obtain $L^{\infty }$ L ∞ decay estimates of weak solutiona.

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