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

Fang Liu

doi:10.3934/cpaa.2018114

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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

Fang Liu ^,

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.

Keywords: Viscosity solution, existence, comparison principle, infinity Laplacian, Lipschitz estimate.

Mathematics Subject Classification: 35D40, 35K20, 35K67, 35Q91.

Citation: Fang Liu. An inhomogeneous evolution equation involving the normalized infinity Laplacian with a transport term. Communications on Pure and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[8]	V. Caselles, J. M. Morel and C. Sbert, An axiomatic approach to image interpolation, IEEE Trans. Image Process, 7 (1998), 376-386.
[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.
[10]	M. G. Crandall and P. Y. Wang, Another way to say caloric, J. Evol. Equ., 3 (2004), 653.
[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.
[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.
[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.
[14]	L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc., 137 (1999).
[15]	L. C. Evans and J. Spruck, Motion of level sets by mean curvature I, J. Differential Geom., 33 (1991), 635-681.
[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.
[17]	L. C. Evans and C. K. Smart, Everywhere differentiability of infinity harmonic functions, Calc. Var. Partial Differential Equations, 42 (2011), 289-299.
[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.
[19]	Y. Giga, Surface Evolution Equations- a Level Set Approach, Birkhäuser, Basel, Switzerland, 2006.
[20]	P. Juutinen and B. Kawohl, On the evolution governed by the infinity Laplacian, Math. Ann., 335 (2006), 819-851.
[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.
[22]	B. Kawohl, Variations on the p-Laplacian, Nonlinear Elliptic Partial Differential Equations, Contemporary Mathematics, 540 (2011), 35-46.
[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.
[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.
[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).
[26]	F. Liu and X. P. Yang, Solutions to an inhomogeneous equation involving infinity-Laplacian, Nonlinear Analysis: Theory, Methods & Applications, 75 (2012), 5693-5701.
[27]	F. Liu and F. Jiang, Parabolic biased infinity Laplacian equation related to the biased tugof-war, Advanced Nonlinear Studies, accepted.
[28]	G. Lu and P. Wang, Infinity Laplace equation with non-trivial right-hand side, Electr. J. Diff. Equ., 77 (2010), 1-12.
[29]	G. Lu and P. Wang, Inhomogeneous infinity Laplace equation, Advances in Mathematics, 217 (2008), 1838-1868.
[30]	G. Lu and P. Wang, A PDE perspective of the normalized infinity Laplacian, Comm. Partial Differential Equations, 33 (2008), 1788-1817.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[8]	V. Caselles, J. M. Morel and C. Sbert, An axiomatic approach to image interpolation, IEEE Trans. Image Process, 7 (1998), 376-386.
[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.
[10]	M. G. Crandall and P. Y. Wang, Another way to say caloric, J. Evol. Equ., 3 (2004), 653.
[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.
[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.
[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.
[14]	L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc., 137 (1999).
[15]	L. C. Evans and J. Spruck, Motion of level sets by mean curvature I, J. Differential Geom., 33 (1991), 635-681.
[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.
[17]	L. C. Evans and C. K. Smart, Everywhere differentiability of infinity harmonic functions, Calc. Var. Partial Differential Equations, 42 (2011), 289-299.
[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.
[19]	Y. Giga, Surface Evolution Equations- a Level Set Approach, Birkhäuser, Basel, Switzerland, 2006.
[20]	P. Juutinen and B. Kawohl, On the evolution governed by the infinity Laplacian, Math. Ann., 335 (2006), 819-851.
[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.
[22]	B. Kawohl, Variations on the p-Laplacian, Nonlinear Elliptic Partial Differential Equations, Contemporary Mathematics, 540 (2011), 35-46.
[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.
[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.
[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).
[26]	F. Liu and X. P. Yang, Solutions to an inhomogeneous equation involving infinity-Laplacian, Nonlinear Analysis: Theory, Methods & Applications, 75 (2012), 5693-5701.
[27]	F. Liu and F. Jiang, Parabolic biased infinity Laplacian equation related to the biased tugof-war, Advanced Nonlinear Studies, accepted.
[28]	G. Lu and P. Wang, Infinity Laplace equation with non-trivial right-hand side, Electr. J. Diff. Equ., 77 (2010), 1-12.
[29]	G. Lu and P. Wang, Inhomogeneous infinity Laplace equation, Advances in Mathematics, 217 (2008), 1838-1868.
[30]	G. Lu and P. Wang, A PDE perspective of the normalized infinity Laplacian, Comm. Partial Differential Equations, 33 (2008), 1788-1817.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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