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November  2018, 17(6): 2395-2421. doi: 10.3934/cpaa.2018114

## An inhomogeneous evolution equation involving the normalized infinity Laplacian with a transport term

 Department of Mathematics, School of Science, Nanjing University of Science & Technology, Nanjing 210094, Jiangsu, China

Received  August 2017 Revised  January 2018 Published  June 2018

Fund Project: This work is supported by the National Natural Science Foundation of China, No.11501292

In this paper, we prove the uniqueness and stability of viscosity solutions of the following initial-boundary problem related to the random game named tug-of-war with a transport term
 $\left\{ \begin{array}{*{35}{l}} {{u}_{t}}-\Delta _{\infty }^{N}u-\langle \xi ,Du\rangle = f(x,t),\ \ \ \ \ \ \text{in}\ \ {{Q}_{T}}, \\ u = g,\ \ \ \ \ \ \ \ \text{on}\ \ \ \ \ {{\partial }_{p}}{{Q}_{T}}, \\\end{array} \right.$
where
 $\Delta _{\infty }^{N}u = \frac{1}{{{\left| Du \right|}^{2}}}\sum\limits_{i,j = 1}^{n}{{{u}_{{{x}_{i}}}}}{{u}_{{{x}_{j}}}}{{u}_{{{x}_{i}}{{x}_{j}}}}$
denotes the normalized infinity Laplacian,
 $ξ: Q_T\to R^n$
is a continuous vector field,
 $f$
and
 $g$
are continuous. When
 $ξ$
is a fixed field and the inhomogeneous term
 $f$
is constant, the existence is obtained by the approximate procedure. When
 $f$
and
 $ξ$
are Lipschitz continuous, we also establish the Lipschitz continuity of the viscosity solutions. Furthermore we establish the comparison principle of the generalized equation with the first order term with initial-boundary condition
 ${u_t}(x,t) -Δ _∞ ^N u (x,t) -H(x,t,Du(x,t)) = f(x,t),$
where
 $H(x,t,p):Q_T× R^n\to R$
is continuous,
 $H(x,t,0) = 0$
and grows at most linearly at infinity with respect to the variable
 $p$
. And the existence result is also obtained when
 $H(x,t,p) = H(p)$
and
 $f$
is constant for the generalized equation.
Citation: Fang Liu. An inhomogeneous evolution equation involving the normalized infinity Laplacian with a transport term. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2395-2421. doi: 10.3934/cpaa.2018114
##### 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. Google Scholar [2] G. Akagi and K. Suzuki, On a certain degenerate parabolic equation associated with the infinity Laplacian, Disc. Cont. Dyna. Sys. supplement, (2007), 18-27. Google Scholar [3] G. Akagi and K. Suzuki, Existence and uniqueness of viscosity solutions for a degenerate parabolic equation associated with the infinity Laplacian, Calc. Var. Partial Differential Equations, 31 (2008), 457-471. Google Scholar [4] G. Akagi, P. Juutinen and R. Kajikiya, Asymptotic behavior of viscosity solutions for a degenerate parabolic equation associated with the infinity Laplacian, Math. Ann., 343 (2009), 921-953. Google Scholar [5] S. N. Armstrong, C. K. Smart and S. J. Somersille, An infinity Laplace equation with gradient term and mixed boundary conditions, Proc. Amer. Math. Soc., 139 (2011), 1763-1776. Google Scholar [6] G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc., 41 (2004), 439-505. Google Scholar [7] E. N. Barron, L. C. Evans and R. Jensen, The infinity Laplacian, Aronsson's equation and their generalizations, Trans. Amer. Math. Soc., 360 (2008), 77-101. Google Scholar [8] V. Caselles, J. M. Morel and C. Sbert, An axiomatic approach to image interpolation, IEEE Trans. Image Process, 7 (1998), 376-386. Google Scholar [9] M. G. Crandall, L. C. Evans and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. Partial Differential Equations, 13 (2001), 123-139. Google Scholar [10] M. G. Crandall and P. Y. Wang, Another way to say caloric, J. Evol. Equ., 3 (2004), 653. Google Scholar [11] M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second-order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67. Google Scholar [12] K. Does, An evolution equation involving the normalized p−Laplacian, Comm. Pure Appl. Anal., 10 (2011), 361–396. Dissertation under the same title, university of Cologne, 2009.Google Scholar [13] A. Elmoataz, M. Toutain and D. Tenbrinck, On the p-Laplacian and ∞-Laplacian on graphs with applications in image and data processing, SIAM J. Imaging Sciences, 8 (2015), 2412-2451. Google Scholar [14] L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc., 137 (1999). Google Scholar [15] L. C. Evans and J. Spruck, Motion of level sets by mean curvature I, J. Differential Geom., 33 (1991), 635-681. Google Scholar [16] L. C. Evans and O. Savin, ${{C}^{1,\alpha }}$ regularity for infinity harmonic functions in two dimensions, Calc. Var. Partial Differential Equations, 32 (2008), 325-347. Google Scholar [17] L. C. Evans and C. K. Smart, Everywhere differentiability of infinity harmonic functions, Calc. Var. Partial Differential Equations, 42 (2011), 289-299. Google Scholar [18] 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. Google Scholar [19] Y. Giga, Surface Evolution Equations- a Level Set Approach, Birkhäuser, Basel, Switzerland, 2006.Google Scholar [20] P. Juutinen and B. Kawohl, On the evolution governed by the infinity Laplacian, Math. Ann., 335 (2006), 819-851. Google Scholar [21] B. Kawohl, J. Manfredi and M. Parviainen, Solutions of nonlinear PDEs in the sense of averages, J. Math. Pure Appl., 97 (2012), 173-188. Google Scholar [22] B. Kawohl, Variations on the p-Laplacian, Nonlinear Elliptic Partial Differential Equations, Contemporary Mathematics, 540 (2011), 35-46. Google Scholar [23] R. Lpez-Soriano, Jos C. Navarro-Climent and Julio D. Rossi, The infinity Laplacian with a transport term, J. Math. Anal. Appl., 398 (2013), 752-765. Google Scholar [24] P. Laurencot and C. Stinner, Refined asymptotics for the infinite heat equation with homogeneous Dirichlet boundary conditions, Comm. Partial Differential Equations, 36 (2010), 532-546. Google Scholar [25] O. A. Ladyženskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, R. I. (1968).Google Scholar [26] F. Liu and X. P. Yang, Solutions to an inhomogeneous equation involving infinity-Laplacian, Nonlinear Analysis: Theory, Methods & Applications, 75 (2012), 5693-5701. Google Scholar [27] F. Liu and F. Jiang, Parabolic biased infinity Laplacian equation related to the biased tugof-war, Advanced Nonlinear Studies, accepted.Google Scholar [28] G. Lu and P. Wang, Infinity Laplace equation with non-trivial right-hand side, Electr. J. Diff. Equ., 77 (2010), 1-12. Google Scholar [29] G. Lu and P. Wang, Inhomogeneous infinity Laplace equation, Advances in Mathematics, 217 (2008), 1838-1868. Google Scholar [30] G. Lu and P. Wang, A PDE perspective of the normalized infinity Laplacian, Comm. Partial Differential Equations, 33 (2008), 1788-1817. Google Scholar [31] J. J. Manfredi, M. Parviainen and J. D. Rossi, Dynamic programming principle for tug-of-war games with noise, ESAIM Control Optim. Calc. Var. COCV, 18 (2012), 81-90. Google Scholar [32] J. J. Manfredi, M. Parviainen and J. D. Rossi, On the definition and properties of p-harmonious functions, Ann. Sc. Norm. Super. Pisa CI. Sci., 11 (2012), 215-241. Google Scholar [33] S. Ohnuma and K. Sato, Singular degenerate parabolic equations with applications to the p-Laplace diffusion equation, Comm. Partial Differential Equations, 22 (1997), 381-411. Google Scholar [34] S. Patrizi, The principal eigenvalue of the ∞-Laplacian with the Neumann boundary condition, ESAIM: Control, Optimisation and Calculus of Variations, 17 (2011), 575-601. Google Scholar [35] Y. Peres, O. Schramm, S. Sheffield and D. Wilson, Tug of war and the infinity Laplacian, J. Amer. Math. Soc., 22 (2009), 167-210. Google Scholar [36] Y. Peres and S. Sheffield, Tug-of-war with noise: a game theoretic view of the p-Laplacian, Duke Math. J., 145 (2008), 91-120. Google Scholar [37] Y. Peres, G. Pete and S. Somersille, Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones, Calc. Var. Partial Differential Equations, 38 (2010), 541-564. Google Scholar [38] M. Portilheiro and J. L. Vazquez, Degenerate homogeneous parabolic equations associated with the infinity-Laplacian, Calc. Var. Partial Differential Equations, 31 (2012), 457-471. Google Scholar [39] M. Portilheiro and J. L. Vazquez, A porous medium equation involving the infinity-Laplacian, Viscosity solutions and asymptotic behaviour,, Comm. Partial Differential Equations, 37 (2012), 753-793. Google Scholar

show all references

##### 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. Google Scholar [2] G. Akagi and K. Suzuki, On a certain degenerate parabolic equation associated with the infinity Laplacian, Disc. Cont. Dyna. Sys. supplement, (2007), 18-27. Google Scholar [3] G. Akagi and K. Suzuki, Existence and uniqueness of viscosity solutions for a degenerate parabolic equation associated with the infinity Laplacian, Calc. Var. Partial Differential Equations, 31 (2008), 457-471. Google Scholar [4] G. Akagi, P. Juutinen and R. Kajikiya, Asymptotic behavior of viscosity solutions for a degenerate parabolic equation associated with the infinity Laplacian, Math. Ann., 343 (2009), 921-953. Google Scholar [5] S. N. Armstrong, C. K. Smart and S. J. Somersille, An infinity Laplace equation with gradient term and mixed boundary conditions, Proc. Amer. Math. Soc., 139 (2011), 1763-1776. Google Scholar [6] G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc., 41 (2004), 439-505. Google Scholar [7] E. N. Barron, L. C. Evans and R. Jensen, The infinity Laplacian, Aronsson's equation and their generalizations, Trans. Amer. Math. Soc., 360 (2008), 77-101. Google Scholar [8] V. Caselles, J. M. Morel and C. Sbert, An axiomatic approach to image interpolation, IEEE Trans. Image Process, 7 (1998), 376-386. Google Scholar [9] M. G. Crandall, L. C. Evans and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. Partial Differential Equations, 13 (2001), 123-139. Google Scholar [10] M. G. Crandall and P. Y. Wang, Another way to say caloric, J. Evol. Equ., 3 (2004), 653. Google Scholar [11] M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second-order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67. Google Scholar [12] K. Does, An evolution equation involving the normalized p−Laplacian, Comm. Pure Appl. Anal., 10 (2011), 361–396. Dissertation under the same title, university of Cologne, 2009.Google Scholar [13] A. Elmoataz, M. Toutain and D. Tenbrinck, On the p-Laplacian and ∞-Laplacian on graphs with applications in image and data processing, SIAM J. Imaging Sciences, 8 (2015), 2412-2451. Google Scholar [14] L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc., 137 (1999). Google Scholar [15] L. C. Evans and J. Spruck, Motion of level sets by mean curvature I, J. Differential Geom., 33 (1991), 635-681. Google Scholar [16] L. C. Evans and O. Savin, ${{C}^{1,\alpha }}$ regularity for infinity harmonic functions in two dimensions, Calc. Var. Partial Differential Equations, 32 (2008), 325-347. Google Scholar [17] L. C. Evans and C. K. Smart, Everywhere differentiability of infinity harmonic functions, Calc. Var. Partial Differential Equations, 42 (2011), 289-299. Google Scholar [18] 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. Google Scholar [19] Y. Giga, Surface Evolution Equations- a Level Set Approach, Birkhäuser, Basel, Switzerland, 2006.Google Scholar [20] P. Juutinen and B. Kawohl, On the evolution governed by the infinity Laplacian, Math. Ann., 335 (2006), 819-851. Google Scholar [21] B. Kawohl, J. Manfredi and M. Parviainen, Solutions of nonlinear PDEs in the sense of averages, J. Math. Pure Appl., 97 (2012), 173-188. Google Scholar [22] B. Kawohl, Variations on the p-Laplacian, Nonlinear Elliptic Partial Differential Equations, Contemporary Mathematics, 540 (2011), 35-46. Google Scholar [23] R. Lpez-Soriano, Jos C. Navarro-Climent and Julio D. Rossi, The infinity Laplacian with a transport term, J. Math. Anal. Appl., 398 (2013), 752-765. Google Scholar [24] P. Laurencot and C. Stinner, Refined asymptotics for the infinite heat equation with homogeneous Dirichlet boundary conditions, Comm. Partial Differential Equations, 36 (2010), 532-546. Google Scholar [25] O. A. Ladyženskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, R. I. (1968).Google Scholar [26] F. Liu and X. P. Yang, Solutions to an inhomogeneous equation involving infinity-Laplacian, Nonlinear Analysis: Theory, Methods & Applications, 75 (2012), 5693-5701. Google Scholar [27] F. Liu and F. Jiang, Parabolic biased infinity Laplacian equation related to the biased tugof-war, Advanced Nonlinear Studies, accepted.Google Scholar [28] G. Lu and P. Wang, Infinity Laplace equation with non-trivial right-hand side, Electr. J. Diff. Equ., 77 (2010), 1-12. Google Scholar [29] G. Lu and P. Wang, Inhomogeneous infinity Laplace equation, Advances in Mathematics, 217 (2008), 1838-1868. Google Scholar [30] G. Lu and P. Wang, A PDE perspective of the normalized infinity Laplacian, Comm. Partial Differential Equations, 33 (2008), 1788-1817. Google Scholar [31] J. J. Manfredi, M. Parviainen and J. D. Rossi, Dynamic programming principle for tug-of-war games with noise, ESAIM Control Optim. Calc. Var. COCV, 18 (2012), 81-90. Google Scholar [32] J. J. Manfredi, M. Parviainen and J. D. Rossi, On the definition and properties of p-harmonious functions, Ann. Sc. Norm. Super. Pisa CI. Sci., 11 (2012), 215-241. Google Scholar [33] S. Ohnuma and K. Sato, Singular degenerate parabolic equations with applications to the p-Laplace diffusion equation, Comm. Partial Differential Equations, 22 (1997), 381-411. Google Scholar [34] S. Patrizi, The principal eigenvalue of the ∞-Laplacian with the Neumann boundary condition, ESAIM: Control, Optimisation and Calculus of Variations, 17 (2011), 575-601. Google Scholar [35] Y. Peres, O. Schramm, S. Sheffield and D. Wilson, Tug of war and the infinity Laplacian, J. Amer. Math. Soc., 22 (2009), 167-210. Google Scholar [36] Y. Peres and S. Sheffield, Tug-of-war with noise: a game theoretic view of the p-Laplacian, Duke Math. J., 145 (2008), 91-120. Google Scholar [37] Y. Peres, G. Pete and S. Somersille, Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones, Calc. Var. Partial Differential Equations, 38 (2010), 541-564. Google Scholar [38] M. Portilheiro and J. L. Vazquez, Degenerate homogeneous parabolic equations associated with the infinity-Laplacian, Calc. Var. Partial Differential Equations, 31 (2012), 457-471. Google Scholar [39] M. Portilheiro and J. L. Vazquez, A porous medium equation involving the infinity-Laplacian, Viscosity solutions and asymptotic behaviour,, Comm. Partial Differential Equations, 37 (2012), 753-793. Google Scholar
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