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Optimal reaction exponent for some qualitative properties of solutions to the $p$-heat equation
Statistical exponential formulas for homogeneous diffusion
1. | Department of Mathematics, Sewanee: The University of the South, Sewanee, TN 37383 |
References:
[1] |
G. Akagi, P. Juutinen and R. Kajikiya, Asymptotic behavior of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian,, \emph{Math. Ann.}, 343 (2009), 921.
doi: 10.1007/s00208-008-0297-1. |
[2] |
G. Akagi and K. Suzuki, Existence and uniqueness of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian,, \emph{Calc. Var. Partial Differential Equations}, 31 (2008), 457.
doi: 10.1007/s00526-007-0117-6. |
[3] |
L. Alvarez, F. Guichard, P.-L. Lions and J.-M. Morel, Axioms and fundamental equations of image processing,, \emph{Arch. Rational Mech. Anal.}, 123 (1993), 199.
doi: 10.1007/BF00375127. |
[4] |
M. Bardi, M. G. Crandall, L. C. Evans, H. M. Soner and P. E. Souganidis, Viscosity solutions and applications, vol. 1660 of Lecture Notes in Mathematics,, Springer-Verlag, (1997), 12.
|
[5] |
G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations,, \emph{Asymptotic Anal.}, 4 (1991), 271.
|
[6] |
G. Barles and C. Georgelin, A simple proof of convergence for an approximation scheme for computing motions by mean curvature,, \emph{SIAM J. Numer. Anal.}, 32 (1995), 484.
doi: 10.1137/0732020. |
[7] |
F. Catté, F. Dibos and G. Koepfler, A morphological scheme for mean curvature motion and applications to anisotropic diffusion and motion of level sets,, \emph{SIAM J. Numer. Anal.}, 32 (1995), 1895.
doi: 10.1137/0732085. |
[8] |
M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces,, \emph{Amer. J. Math.}, 93 (1971), 265.
|
[9] |
M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 27 (1992), 1.
doi: 10.1090/S0273-0979-1992-00266-5. |
[10] |
E. DiBenedetto, Degenerate Parabolic Equations,, Universitext, (1993).
doi: 10.1007/978-1-4612-0895-2. |
[11] |
L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I,, \emph{J. Differential Geom.}, 33 (1991), 635.
|
[12] |
L. C. Evans and J. Spruck, Motion of level sets by mean curvature. II,, \emph{Trans. Amer. Math. Soc.}, 330 (1992), 321.
doi: 10.2307/2154167. |
[13] |
L. C. Evans and J. Spruck, Motion of level sets by mean curvature. III,, \emph{J. Geom. Anal.}, 2 (1992), 121.
doi: 10.1007/BF02921385. |
[14] |
L. C. Evans, Convergence of an algorithm for mean curvature motion,, \emph{Indiana Univ. Math. J.}, 42 (1993), 533.
doi: 10.1512/iumj.1993.42.42024. |
[15] |
L. C. Evans and J. Spruck, Motion of level sets by mean curvature. IV,, \emph{J. Geom. Anal.}, 5 (1995), 77.
doi: 10.1007/BF02926443. |
[16] |
Y. Giga, S. Goto, H. Ishii and M.-H. Sato, Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains,, \emph{Indiana Univ. Math. J.}, 40 (1991), 443.
doi: 10.1512/iumj.1991.40.40023. |
[17] |
Y. Giga, Surface Evolution Equations, vol. 99 of Monographs in Mathematics,, Birkh\, (2006).
|
[18] |
J. A. Goldstein, Semigroups of Linear Operators and Applications,, Oxford Mathematical Monographs, (1985).
|
[19] |
A. Grigor'yan, Heat Kernel and Analysis on Manifolds, vol. 47 of AMS/IP Studies in Advanced Mathematics,, American Mathematical Society, (2009).
|
[20] |
D. Hartenstine and M. Rudd, Asymptotic statistical characterizations of $p$-harmonic functions of two variables,, \emph{Rocky Mountain J. Math.}, 41 (2011), 493.
doi: 10.1216/RMJ-2011-41-2-493. |
[21] |
D. Hartenstine and M. Rudd, Statistical functional equations and $p$-harmonious functions,, \emph{Adv. Nonlinear Stud.}, 13 (2013), 191.
|
[22] |
J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations,, Oxford Mathematical Monographs, (1993).
|
[23] |
T. Ilmanen, P. Sternberg and W. P. Ziemer, Equilibrium solutions to generalized motion by mean curvature,, \emph{J. Geom. Anal.}, 8 (1998), 845.
doi: 10.1007/BF02922673. |
[24] |
H. Ishii, G. E. Pires and P. E. Souganidis, Threshold dynamics type approximation schemes for propagating fronts,, \emph{J. Math. Soc. Japan}, 51 (1999), 267.
doi: 10.2969/jmsj/05120267. |
[25] |
V. Julin and P. Juutinen, A new proof for the equivalence of weak and viscosity solutions for the $p$-Laplace equation,, \emph{Comm. Partial Differential Equations}, 37 (2012), 934.
doi: 10.1080/03605302.2011.615878. |
[26] |
P. Juutinen, $p$-harmonic approximation of functions of least gradient,, \emph{Indiana Univ. Math. J.}, 54 (2005), 1015.
doi: 10.1512/iumj.2005.54.2658. |
[27] |
P. Juutinen and B. Kawohl, On the evolution governed by the infinity Laplacian,, \emph{Math. Ann.}, 335 (2006), 819.
doi: 10.1007/s00208-006-0766-3. |
[28] |
P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation,, \emph{SIAM J. Math. Anal.}, 33 (2001), 699.
doi: 10.1137/S0036141000372179. |
[29] |
B. Kawohl and N. Kutev, Comparison principle for viscosity solutions of fully nonlinear, degenerate elliptic equations,, \emph{Comm. Partial Differential Equations}, 32 (2007), 1209.
doi: 10.1080/03605300601113043. |
[30] |
B. Kawohl, Variations on the $p$-Laplacian,, in \emph{Nonlinear elliptic partial differential equations}, (2011), 35.
doi: 10.1090/conm/540/10657. |
[31] |
B. Kawohl, J. Manfredi and M. Parviainen, Solutions of nonlinear PDEs in the sense of averages,, \emph{J. Math. Pures Appl.}, 97 (2012), 173.
doi: 10.1016/j.matpur.2011.07.001. |
[32] |
B. Kawohl and N. Kutev, Comparison principle and Lipschitz regularity for viscosity solutions of some classes of nonlinear partial differential equations,, \emph{Funkcial. Ekvac.}, 43 (2000), 241.
|
[33] |
R. V. Kohn and S. Serfaty, A deterministic-control-based approach to motion by curvature,, \emph{Comm. Pure Appl. Math.}, 59 (2006), 344.
doi: 10.1002/cpa.20101. |
[34] |
G. F. Lawler, Random Walk and the Heat Equation, vol. 55 of Student Mathematical Library,, American Mathematical Society, (2010).
|
[35] |
P. D. Lax, Functional Analysis,, Pure and Applied Mathematics, (2002).
|
[36] |
E. Le Gruyer, On absolutely minimizing Lipschitz extensions and PDE $\Delta_\infty(u)=0$,, \emph{NoDEA Nonlinear Differential Equations Appl.}, 14 (2007), 29.
doi: 10.1007/s00030-006-4030-z. |
[37] |
E. Le Gruyer and J. C. Archer, Harmonious extensions,, \emph{SIAM J. Math. Anal.}, 29 (1998), 279.
doi: 10.1137/S0036141095294067. |
[38] |
G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co. Inc., (1996).
doi: 10.1142/3302. |
[39] |
P. Lindqvist, Notes on the $p$-Laplace equation, vol. 102 of Report, University of Jyväskylä Department of Mathematics and Statistics,, University of Jyv\, (2006).
|
[40] |
J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value characterization for a class of nonlinear parabolic equations related to tug-of-war games,, \emph{SIAM J. Math. Anal.}, 42 (2010), 2058.
doi: 10.1137/100782073. |
[41] |
B. Merriman, J. K. Bence and S. J. Osher, Motion of multiple functions: a level set approach,, \emph{J. Comput. Phys.}, 112 (1994), 334.
doi: 10.1006/jcph.1994.1105. |
[42] |
A. M. Oberman, A convergent monotone difference scheme for motion of level sets by mean curvature,, \emph{Numer. Math.}, 99 (2004), 365.
doi: 10.1007/s00211-004-0566-1. |
[43] |
A. M. Oberman, A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions,, \emph{Math. Comp.}, 74 (2005), 1217.
doi: 10.1090/S0025-5718-04-01688-6. |
[44] |
S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations,, \emph{J. Comput. Phys.}, 79 (1988), 12.
doi: 10.1016/0021-9991(88)90002-2. |
[45] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, vol. 44 of Applied Mathematical Sciences, (1983).
doi: 10.1007/978-1-4612-5561-1. |
[46] |
Y. Peres and S. Sheffield, Tug-of-war with noise: a game-theoretic view of the $p$-Laplacian,, \emph{Duke Math. J.}, 145 (2008), 91.
doi: 10.1215/00127094-2008-048. |
[47] |
S. J. Ruuth and B. Merriman, Convolution-generated motion and generalized Huygens' principles for interface motion,, \emph{SIAM J. Appl. Math.}, 60 (2000), 868.
doi: 10.1137/S003613999833397X. |
[48] |
P. Sternberg and W. P. Ziemer, Generalized motion by curvature with a Dirichlet condition,, \emph{J. Differential Equations}, 114 (1994), 580.
doi: 10.1006/jdeq.1994.1162. |
[49] |
N. T. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, vol. 100 of Cambridge Tracts in Mathematics,, Cambridge University Press, (1992).
|
[50] |
W. P. Ziemer, Weakly Differentiable Functions,, vol. 120 of Graduate Texts in Mathematics, (1989).
doi: 10.1007/978-1-4612-1015-3. |
show all references
References:
[1] |
G. Akagi, P. Juutinen and R. Kajikiya, Asymptotic behavior of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian,, \emph{Math. Ann.}, 343 (2009), 921.
doi: 10.1007/s00208-008-0297-1. |
[2] |
G. Akagi and K. Suzuki, Existence and uniqueness of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian,, \emph{Calc. Var. Partial Differential Equations}, 31 (2008), 457.
doi: 10.1007/s00526-007-0117-6. |
[3] |
L. Alvarez, F. Guichard, P.-L. Lions and J.-M. Morel, Axioms and fundamental equations of image processing,, \emph{Arch. Rational Mech. Anal.}, 123 (1993), 199.
doi: 10.1007/BF00375127. |
[4] |
M. Bardi, M. G. Crandall, L. C. Evans, H. M. Soner and P. E. Souganidis, Viscosity solutions and applications, vol. 1660 of Lecture Notes in Mathematics,, Springer-Verlag, (1997), 12.
|
[5] |
G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations,, \emph{Asymptotic Anal.}, 4 (1991), 271.
|
[6] |
G. Barles and C. Georgelin, A simple proof of convergence for an approximation scheme for computing motions by mean curvature,, \emph{SIAM J. Numer. Anal.}, 32 (1995), 484.
doi: 10.1137/0732020. |
[7] |
F. Catté, F. Dibos and G. Koepfler, A morphological scheme for mean curvature motion and applications to anisotropic diffusion and motion of level sets,, \emph{SIAM J. Numer. Anal.}, 32 (1995), 1895.
doi: 10.1137/0732085. |
[8] |
M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces,, \emph{Amer. J. Math.}, 93 (1971), 265.
|
[9] |
M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 27 (1992), 1.
doi: 10.1090/S0273-0979-1992-00266-5. |
[10] |
E. DiBenedetto, Degenerate Parabolic Equations,, Universitext, (1993).
doi: 10.1007/978-1-4612-0895-2. |
[11] |
L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I,, \emph{J. Differential Geom.}, 33 (1991), 635.
|
[12] |
L. C. Evans and J. Spruck, Motion of level sets by mean curvature. II,, \emph{Trans. Amer. Math. Soc.}, 330 (1992), 321.
doi: 10.2307/2154167. |
[13] |
L. C. Evans and J. Spruck, Motion of level sets by mean curvature. III,, \emph{J. Geom. Anal.}, 2 (1992), 121.
doi: 10.1007/BF02921385. |
[14] |
L. C. Evans, Convergence of an algorithm for mean curvature motion,, \emph{Indiana Univ. Math. J.}, 42 (1993), 533.
doi: 10.1512/iumj.1993.42.42024. |
[15] |
L. C. Evans and J. Spruck, Motion of level sets by mean curvature. IV,, \emph{J. Geom. Anal.}, 5 (1995), 77.
doi: 10.1007/BF02926443. |
[16] |
Y. Giga, S. Goto, H. Ishii and M.-H. Sato, Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains,, \emph{Indiana Univ. Math. J.}, 40 (1991), 443.
doi: 10.1512/iumj.1991.40.40023. |
[17] |
Y. Giga, Surface Evolution Equations, vol. 99 of Monographs in Mathematics,, Birkh\, (2006).
|
[18] |
J. A. Goldstein, Semigroups of Linear Operators and Applications,, Oxford Mathematical Monographs, (1985).
|
[19] |
A. Grigor'yan, Heat Kernel and Analysis on Manifolds, vol. 47 of AMS/IP Studies in Advanced Mathematics,, American Mathematical Society, (2009).
|
[20] |
D. Hartenstine and M. Rudd, Asymptotic statistical characterizations of $p$-harmonic functions of two variables,, \emph{Rocky Mountain J. Math.}, 41 (2011), 493.
doi: 10.1216/RMJ-2011-41-2-493. |
[21] |
D. Hartenstine and M. Rudd, Statistical functional equations and $p$-harmonious functions,, \emph{Adv. Nonlinear Stud.}, 13 (2013), 191.
|
[22] |
J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations,, Oxford Mathematical Monographs, (1993).
|
[23] |
T. Ilmanen, P. Sternberg and W. P. Ziemer, Equilibrium solutions to generalized motion by mean curvature,, \emph{J. Geom. Anal.}, 8 (1998), 845.
doi: 10.1007/BF02922673. |
[24] |
H. Ishii, G. E. Pires and P. E. Souganidis, Threshold dynamics type approximation schemes for propagating fronts,, \emph{J. Math. Soc. Japan}, 51 (1999), 267.
doi: 10.2969/jmsj/05120267. |
[25] |
V. Julin and P. Juutinen, A new proof for the equivalence of weak and viscosity solutions for the $p$-Laplace equation,, \emph{Comm. Partial Differential Equations}, 37 (2012), 934.
doi: 10.1080/03605302.2011.615878. |
[26] |
P. Juutinen, $p$-harmonic approximation of functions of least gradient,, \emph{Indiana Univ. Math. J.}, 54 (2005), 1015.
doi: 10.1512/iumj.2005.54.2658. |
[27] |
P. Juutinen and B. Kawohl, On the evolution governed by the infinity Laplacian,, \emph{Math. Ann.}, 335 (2006), 819.
doi: 10.1007/s00208-006-0766-3. |
[28] |
P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation,, \emph{SIAM J. Math. Anal.}, 33 (2001), 699.
doi: 10.1137/S0036141000372179. |
[29] |
B. Kawohl and N. Kutev, Comparison principle for viscosity solutions of fully nonlinear, degenerate elliptic equations,, \emph{Comm. Partial Differential Equations}, 32 (2007), 1209.
doi: 10.1080/03605300601113043. |
[30] |
B. Kawohl, Variations on the $p$-Laplacian,, in \emph{Nonlinear elliptic partial differential equations}, (2011), 35.
doi: 10.1090/conm/540/10657. |
[31] |
B. Kawohl, J. Manfredi and M. Parviainen, Solutions of nonlinear PDEs in the sense of averages,, \emph{J. Math. Pures Appl.}, 97 (2012), 173.
doi: 10.1016/j.matpur.2011.07.001. |
[32] |
B. Kawohl and N. Kutev, Comparison principle and Lipschitz regularity for viscosity solutions of some classes of nonlinear partial differential equations,, \emph{Funkcial. Ekvac.}, 43 (2000), 241.
|
[33] |
R. V. Kohn and S. Serfaty, A deterministic-control-based approach to motion by curvature,, \emph{Comm. Pure Appl. Math.}, 59 (2006), 344.
doi: 10.1002/cpa.20101. |
[34] |
G. F. Lawler, Random Walk and the Heat Equation, vol. 55 of Student Mathematical Library,, American Mathematical Society, (2010).
|
[35] |
P. D. Lax, Functional Analysis,, Pure and Applied Mathematics, (2002).
|
[36] |
E. Le Gruyer, On absolutely minimizing Lipschitz extensions and PDE $\Delta_\infty(u)=0$,, \emph{NoDEA Nonlinear Differential Equations Appl.}, 14 (2007), 29.
doi: 10.1007/s00030-006-4030-z. |
[37] |
E. Le Gruyer and J. C. Archer, Harmonious extensions,, \emph{SIAM J. Math. Anal.}, 29 (1998), 279.
doi: 10.1137/S0036141095294067. |
[38] |
G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co. Inc., (1996).
doi: 10.1142/3302. |
[39] |
P. Lindqvist, Notes on the $p$-Laplace equation, vol. 102 of Report, University of Jyväskylä Department of Mathematics and Statistics,, University of Jyv\, (2006).
|
[40] |
J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value characterization for a class of nonlinear parabolic equations related to tug-of-war games,, \emph{SIAM J. Math. Anal.}, 42 (2010), 2058.
doi: 10.1137/100782073. |
[41] |
B. Merriman, J. K. Bence and S. J. Osher, Motion of multiple functions: a level set approach,, \emph{J. Comput. Phys.}, 112 (1994), 334.
doi: 10.1006/jcph.1994.1105. |
[42] |
A. M. Oberman, A convergent monotone difference scheme for motion of level sets by mean curvature,, \emph{Numer. Math.}, 99 (2004), 365.
doi: 10.1007/s00211-004-0566-1. |
[43] |
A. M. Oberman, A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions,, \emph{Math. Comp.}, 74 (2005), 1217.
doi: 10.1090/S0025-5718-04-01688-6. |
[44] |
S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations,, \emph{J. Comput. Phys.}, 79 (1988), 12.
doi: 10.1016/0021-9991(88)90002-2. |
[45] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, vol. 44 of Applied Mathematical Sciences, (1983).
doi: 10.1007/978-1-4612-5561-1. |
[46] |
Y. Peres and S. Sheffield, Tug-of-war with noise: a game-theoretic view of the $p$-Laplacian,, \emph{Duke Math. J.}, 145 (2008), 91.
doi: 10.1215/00127094-2008-048. |
[47] |
S. J. Ruuth and B. Merriman, Convolution-generated motion and generalized Huygens' principles for interface motion,, \emph{SIAM J. Appl. Math.}, 60 (2000), 868.
doi: 10.1137/S003613999833397X. |
[48] |
P. Sternberg and W. P. Ziemer, Generalized motion by curvature with a Dirichlet condition,, \emph{J. Differential Equations}, 114 (1994), 580.
doi: 10.1006/jdeq.1994.1162. |
[49] |
N. T. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, vol. 100 of Cambridge Tracts in Mathematics,, Cambridge University Press, (1992).
|
[50] |
W. P. Ziemer, Weakly Differentiable Functions,, vol. 120 of Graduate Texts in Mathematics, (1989).
doi: 10.1007/978-1-4612-1015-3. |
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