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A Liouville type theorem to an extension problem relating to the Heisenberg group
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 |
$\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. $ |
$ \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}}}}$ |
$ ξ: Q_T\to R^n$ |
$ f$ |
$ g$ |
$ ξ$ |
$ f$ |
$ f$ |
$ ξ$ |
${u_t}(x,t) -Δ _∞ ^N u (x,t) -H(x,t,Du(x,t)) = f(x,t),$ |
$ H(x,t,p):Q_T× R^n\to R$ |
$ H(x,t,0) = 0$ |
$ p$ |
$ H(x,t,p) = H(p)$ |
$ f$ |
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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