有偏无穷Laplace方程解的若干估计和性质
|
摘要
本文研究一类有偏无穷Laplace方程,它来源于随机博弈论中的有偏二人零和博弈.本文建立该方程解的各种性质,包括解的梯度估计、非负解u及其梯度模|Du|的Harnack不等式.最后,本文证明非常数的C~2光滑解没有内部临界点.
In this paper, we study a kind of infinity Laplacian equations arising from biased tug-of-war in random games. Various properties of the solutions for such equations, including the gradient estimates, the Harnack inequalities for both nonnegative u and the gradient |Du|, are established. Finally, we prove that there are no interior critical points for non-constant C~2 solutions.
In this paper, we study a kind of infinity Laplacian equations arising from biased tug-of-war in random games. Various properties of the solutions for such equations, including the gradient estimates, the Harnack inequalities for both nonnegative u and the gradient |Du|, are established. Finally, we prove that there are no interior critical points for non-constant C~2 solutions.
引文
1 Peres Y,Pete G,Somersille S.Biased tug-of-war,the biased infinity Laplacian,and comparison with exponential cones.Calc Var Partial Differential Equations,2010,38:541-564
2 Barron E N,Evans L C,Jensen R.The infinity Laplacian,Aronsson’s equation and their generalizations.Trans Amer Math Soc,2008,360:77-101
3 Evans L C.Estimates for smooth absolutely minimizing Lipschitz extensions.Electron J Differential Equations,1993,3:1-10
4 Yu Y.A remark on C2infinity-harmonic functions.Electron J Differential Equations,2006,122:1-4
5 Aronsson G.On the partial differential equation u2xuxx+2uxuyuxy+u2yuyy=0.Ark Mat,1968,7:395-425
6 Savin O.C1regularity for infinity harmonic functions in two dimensions.Arch Ration Mech Anal,2005,176:351-361
7 Evans L C,Savin O.C1,αregularity for infinity harmonic functions in two dimensions.Calc Var Partial Differential Equations,2008,32:325-347
8 Evans L C,Smart C K.Everywhere differentiability of infinity harmonic functions.Calc Var Partial Differential Equations,2011,42:289-299
9 Jiang F,Trudinger N S,Yang X P.On the Dirichlet problem for Monge-Ampére type equations.Calc Var Partial Differential Equations,2014,49:1223-1236
10 Jiang F,Trudinger N S,Xiang N.On the Neumann problem for Monge-Ampére type equations.Canad J Math,2016,68:1334-1361
11 Armstrong S N,Smart C K,Somersille S J.An infinity Laplace equation with gradient term and mixed boundary conditions.Proc Amer Math Soc,2011,139:1763-1776
12 Barles G,Busca J.Existence and comparison results for fully nonlinear degenerate elliptic equations without zerothorder term.Comm Partial Differential Equations,2001,26:2323-2337
13 Liu Q,Schikorra A.General existence of solutions to dynamic programming equations.Commun Pure Appl Anal,2015,14:167-184
14 Peres Y,Schramm O,Sheffield S,et al.Tug-of-war and the infinity Laplacian.J Amer Math Soc,2009,22:167-210
15 Charro F,García Azorero J,Rossi J D.A mixed problem for the infinity Laplacian via tug-of-war games.Calc Var Partial Differential Equations,2009,34:307-320
16 Evans L C,Gangbo W.Differential equations methods for the Monge-Kantorovich mass transfer problem.Mem Amer Math Soc,1999,137:66pp
17 García Azorero J,Manfredi J J,Peral I,et al.The Neumann problem for the∞-Laplacian and the Monge-Kantorovich mass transfer problem.Nonlinear Anal,2007,66:349-366
18 Bonamy C,Le Guyader C.Split Bregman iteration and infinity Laplacian for image decomposition.J Comput Appl Math,2013,240:99-110
19 Caselles V,Morel J M,Sbert C.An axiomatic approach to image interpolation.IEEE Trans Image Process,1998,7:376-386
20 Elion C,Vese L A.An image decomposition model using the total variation and the infinity Laplacian.In:Proceedings of Electronic Imaging 2007.Bellingham:SPIE,2007,64980W
21 Guillot L,Le Guyader C.Extrapolation of vector fields using the infinity Laplacian and with applications to image segmentation.In:International Conference on Scale Space and Variational Methods in Computer Vision.HeidelbergBerlin:Springer,2009,87-99
22 Bhattacharya T.An elementary proof of the Harnack inequality for non-negative infinity-superharmonic functions.Electron J Differential Equations,2001,44:1-8
23 Bhattacharya T.A note on non-negative singular infinity-harmonic functions in the half-space.Rev Mat Complut,2005,18:377-385
24 Jensen R.Uniqueness of Lipschitz extensions:Minimizing the sup norm of the gradient.Arch Ration Mech Anal,1993,123:51-74
25 Lindqvist P,Manfredi J J.The Harnack inequality for∞-harmonic functions.Electron J Differential Equations,1995,4:1-5
26 Bhataacharya T,Mohammed A.On solutions to Dirichlet problems involving the infinity-Laplacian.Adv Calc Var,2011,4:445-487
27 Liu F,Yang X P.Solutions to an inhomogeneous equation involving infinity Laplacian.Nonlinear Anal,2012,75:5693-5701
28 Lu G,Wang P.A PDE perspective of the normalized infinity Laplacian.Comm Partial Differential Equations,2008,33:1788-1817
29 Lu G,Wang P.Inhomogeneous infinity Laplace equation.Adv Math,2008,217:1838-1868
30 Lu G,Wang P.Infinity Laplace equation with non-trivial right-hand side.Electron J Differential Equations,2010,77:1-12
31 López-Soriano R,Navarro-Climent J C,Rossi J D.The infinity Laplacian with a transport term.J Math Anal Appl,2013,398:752-765
32 Gilbarg D,Trudinger N S.Elliptic Partial Differential Equations of Second Order,Reprint of the 1998 Edition.Berlin:Springer-Verlag,2001
33 Llorente J G.Mean value properties and unique continuation.Commun Pure Appl Anal,2015,14:185-199
1)Jiang F, Liu F, Yang X P. Lipschitz estimates for solutions to a porous medium problem involving the infinity Laplacian.Preprint, 2017
2 Barron E N,Evans L C,Jensen R.The infinity Laplacian,Aronsson’s equation and their generalizations.Trans Amer Math Soc,2008,360:77-101
3 Evans L C.Estimates for smooth absolutely minimizing Lipschitz extensions.Electron J Differential Equations,1993,3:1-10
4 Yu Y.A remark on C2infinity-harmonic functions.Electron J Differential Equations,2006,122:1-4
5 Aronsson G.On the partial differential equation u2xuxx+2uxuyuxy+u2yuyy=0.Ark Mat,1968,7:395-425
6 Savin O.C1regularity for infinity harmonic functions in two dimensions.Arch Ration Mech Anal,2005,176:351-361
7 Evans L C,Savin O.C1,αregularity for infinity harmonic functions in two dimensions.Calc Var Partial Differential Equations,2008,32:325-347
8 Evans L C,Smart C K.Everywhere differentiability of infinity harmonic functions.Calc Var Partial Differential Equations,2011,42:289-299
9 Jiang F,Trudinger N S,Yang X P.On the Dirichlet problem for Monge-Ampére type equations.Calc Var Partial Differential Equations,2014,49:1223-1236
10 Jiang F,Trudinger N S,Xiang N.On the Neumann problem for Monge-Ampére type equations.Canad J Math,2016,68:1334-1361
11 Armstrong S N,Smart C K,Somersille S J.An infinity Laplace equation with gradient term and mixed boundary conditions.Proc Amer Math Soc,2011,139:1763-1776
12 Barles G,Busca J.Existence and comparison results for fully nonlinear degenerate elliptic equations without zerothorder term.Comm Partial Differential Equations,2001,26:2323-2337
13 Liu Q,Schikorra A.General existence of solutions to dynamic programming equations.Commun Pure Appl Anal,2015,14:167-184
14 Peres Y,Schramm O,Sheffield S,et al.Tug-of-war and the infinity Laplacian.J Amer Math Soc,2009,22:167-210
15 Charro F,García Azorero J,Rossi J D.A mixed problem for the infinity Laplacian via tug-of-war games.Calc Var Partial Differential Equations,2009,34:307-320
16 Evans L C,Gangbo W.Differential equations methods for the Monge-Kantorovich mass transfer problem.Mem Amer Math Soc,1999,137:66pp
17 García Azorero J,Manfredi J J,Peral I,et al.The Neumann problem for the∞-Laplacian and the Monge-Kantorovich mass transfer problem.Nonlinear Anal,2007,66:349-366
18 Bonamy C,Le Guyader C.Split Bregman iteration and infinity Laplacian for image decomposition.J Comput Appl Math,2013,240:99-110
19 Caselles V,Morel J M,Sbert C.An axiomatic approach to image interpolation.IEEE Trans Image Process,1998,7:376-386
20 Elion C,Vese L A.An image decomposition model using the total variation and the infinity Laplacian.In:Proceedings of Electronic Imaging 2007.Bellingham:SPIE,2007,64980W
21 Guillot L,Le Guyader C.Extrapolation of vector fields using the infinity Laplacian and with applications to image segmentation.In:International Conference on Scale Space and Variational Methods in Computer Vision.HeidelbergBerlin:Springer,2009,87-99
22 Bhattacharya T.An elementary proof of the Harnack inequality for non-negative infinity-superharmonic functions.Electron J Differential Equations,2001,44:1-8
23 Bhattacharya T.A note on non-negative singular infinity-harmonic functions in the half-space.Rev Mat Complut,2005,18:377-385
24 Jensen R.Uniqueness of Lipschitz extensions:Minimizing the sup norm of the gradient.Arch Ration Mech Anal,1993,123:51-74
25 Lindqvist P,Manfredi J J.The Harnack inequality for∞-harmonic functions.Electron J Differential Equations,1995,4:1-5
26 Bhataacharya T,Mohammed A.On solutions to Dirichlet problems involving the infinity-Laplacian.Adv Calc Var,2011,4:445-487
27 Liu F,Yang X P.Solutions to an inhomogeneous equation involving infinity Laplacian.Nonlinear Anal,2012,75:5693-5701
28 Lu G,Wang P.A PDE perspective of the normalized infinity Laplacian.Comm Partial Differential Equations,2008,33:1788-1817
29 Lu G,Wang P.Inhomogeneous infinity Laplace equation.Adv Math,2008,217:1838-1868
30 Lu G,Wang P.Infinity Laplace equation with non-trivial right-hand side.Electron J Differential Equations,2010,77:1-12
31 López-Soriano R,Navarro-Climent J C,Rossi J D.The infinity Laplacian with a transport term.J Math Anal Appl,2013,398:752-765
32 Gilbarg D,Trudinger N S.Elliptic Partial Differential Equations of Second Order,Reprint of the 1998 Edition.Berlin:Springer-Verlag,2001
33 Llorente J G.Mean value properties and unique continuation.Commun Pure Appl Anal,2015,14:185-199
1)Jiang F, Liu F, Yang X P. Lipschitz estimates for solutions to a porous medium problem involving the infinity Laplacian.Preprint, 2017