Comparison principle to the infinity Laplacian equation with lower term

Volume 6, Issue 5, October 2021     |     PP. 51-62      |     PDF (395 K)    |     Pub. Date: November 9, 2021
DOI: 10.54647/mathematics11292    62 Downloads     2620 Views  

Author(s)

Cuicui Li, School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, People’s Republic of China
Fang Liu, School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, People’s Republic of China

Abstract
In this paper, we establish the comparison principles via the perturbation method for the equation in , where is a highly degenerate and h- homogeneous operator associated with the infinity Laplacian. Based on the comparison principle, we obtain the uniqueness of the viscosity solution to the Dirichlet problem

where During this procedure, we also establish a stability result of the viscosity solution to the inhomogeneous equation.

Keywords
infinity Laplacian; viscosity solutions; comparison principle; stability.

Cite this paper
Cuicui Li, Fang Liu, Comparison principle to the infinity Laplacian equation with lower term , SCIREA Journal of Mathematics. Volume 6, Issue 5, October 2021 | PP. 51-62. 10.54647/mathematics11292

References

[ 1 ] E. Abderrahim, D. Xavier, L. Zakariaa and L. Olivier, Nonlocal infinity Laplacian equation on graphs with applications in image processing and machine learning, Mathematics and Computers in Simulation, 102(2014), 153-163.
[ 2 ] G. Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat. 6(1967), 551-561.
[ 3 ] G. Aronsson, M. G. Crandall, P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc. 41(2004), 439-505.
[ 4 ] S. Armstrong, C. Smart, An easy proof Jensen’s theorem on the uniqueness of infinity harmonic functions. Calc. Var. Partial Differ. Equ. 37(2010), 381-384.
[ 5 ] T. Bhattacharya, A. Mohammed, On solutions to Dirichlet problems involving the infinity-Laplacian, Adv. Calc. Var. 4(2011), 445-487.
[ 6 ] V. Caselles, J. M. Morel and C. Sbert, An axiomatic approach to image interpolation, IEEE Trans. Image Process, 7(1998), 376-386.
[ 7 ] M. G. Crandall, A visit with the ∞-Laplace equation, in Calculus of Variations and Nonlinear Partial Differential Equations, Lecture Notes in Math. 1927, pp. 75-122, Springer, Berlin, 2008.
[ 8 ] M. G. Crandall, L. C. Evans and R. F. Gariepy, Optimal Lipschitz extensions and the infinity-Laplacian, Calc. Var. Partial Differ. Equ. 13(2001), 123-139.
[ 9 ] M. G. Crandall, P. L. Lions, Viscosity solutions and Hamilton-Jacobi equations. Trans. Am. Math. Soc. 277(1983), 1-42.
[ 10 ] M. G. Crandall, L. C. Evans, P. L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 282(1984), 487-502.
[ 11 ] M. G. Crandall, H. Ishii, P. L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N. S.) 27(1992), 1-67.
[ 12 ] L. C. Evansand W.Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc., 137 (1999) no.653, viii+66 pp.
[ 13 ] A. Elmoataz, M. Toutain, D. Tenbrinck, On the p-Laplacian and ∞-Laplacian on graphs with applications in image and data processing, SIAM J. Imaging Sciences, 8(4) (2015), 2412-2451.
[ 14 ] J. Garcia-Azorero, J. J. Manfredi, I. Peral and J. D. Rossi, The Neumann problem for the ∞−Laplacian and the Monge-Kantorovich mass transfer problem, Nonlinear Analysis: Theory Methods & Applications, 66 (2007), 349–366.
[ 15 ] F. Liu, An inhomogeneous evolution equation involving the normalized infinity Laplacian with a transport term, Commun Pure Appl Anal, 17(6) (2018), 2395-2421.
[ 16 ] F. Liu, F. Jiang, Parabolic Biased Infinity Laplacian Equation Related to the Biased Tug-of-War, Advanced Nonlinear Studies, 19(1) (2019), 89-112.
[ 17 ] F. Liu, X.P. Yang, Solutions to an inhomogeneous equation involving infinity-Laplacian, Nonlinear Analysis: Theory, Methods & Applications, 75 (2012), 5693-5701.
[ 18 ] R. López-Soriano, J. C. Navarro-Climent and J. D. Rossi, The infinity Laplacian with a transport term. J. Math. Anal. Appl. 398(2013), 752-765.
[ 19 ] G. Lu, P. Wang, A PDE perspective of the normalized infinity Laplacian, Comm. Part. Diff. Eqns. 33(10) (2008), 1788-1817.
[ 20 ] G. Lu, P. Wang, Inhomogeneous infinity Laplace equation. Adv. Math. 217(2008), 1838-1868.
[ 21 ] K. Nyström, M. Parviainen, Tug-of-war, market manipulation, and option pricing. Math. Finance, 27(2017), 279-312.
[ 22 ] Y. Peres, G. Pete, S. Somersille, Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones, Calc. Var. PDE. 38(3-4) (2010), 541-564.
[ 23 ] Y. Peres, O. Schramm, S. Sheffield, D. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc. 22(1) (2009), 167-210.
[ 24 ] J. D. Rossi, Tug-of-war games and PDEs, Proc. Roy. Soc. Edinburgh Sect. A 141(2) (2011), 319-369.