American Institute of Mathematical Sciences

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January  2015, 14(1): 269-284. doi: 10.3934/cpaa.2015.14.269

Statistical exponential formulas for homogeneous diffusion

Matthew B. Rudd 1,

1. 

Department of Mathematics, Sewanee: The University of the South, Sewanee, TN 37383

Received  March 2014 Revised  April 2014 Published  September 2014

Let $\Delta^{1}_{p}$ denote the $1$-homogeneous $p$-Laplacian, for $1 \leq p \leq \infty$. This paper proves that the unique bounded, continuous viscosity solution $u$ of the Cauchy problem \begin{eqnarray} u_{t} - ( \frac{p}{ N + p - 2 } ) \Delta^{1}_{p} u = 0 \quad \mbox{for} \quad x \in R^N \quad \mbox{and} \quad t > 0 , \\ \\ u(\cdot,0) = u_0 \in BUC(R^N). \end{eqnarray} is given by the exponential formula \begin{eqnarray} u(t) := \lim_{n \to \infty}{ ( M^{t/n}_{p} )^{n} u_{0} } \ , \end{eqnarray} where the statistical operator $M^h_p \colon BUC( R^{N} ) \to BUC( R^{N} )$ is defined by \begin{eqnarray} (M^{h}_{p} \varphi)(x) := (1-q) median_{\partial B(x,\sqrt{2h})}{ \{ \varphi \} } + q \int_{\partial B(x,\sqrt{2h})}{ \varphi ds } \end{eqnarray} when $1 \leq p \leq 2$, with $q := \frac{ N ( p - 1 ) }{ N + p - 2 }$, and by \begin{eqnarray} (M^{h}_{p} \varphi )(x) := ( 1 - q ) midrange_{\partial B(x,\sqrt{2h})}{ \{ \varphi\} } + q \int_{\partial B(x,\sqrt{2h})}{ \varphi ds } \end{eqnarray} when $p \geq 2$, with $q = \frac{ N }{ N + p - 2 }$. Possible extensions to problems with Dirichlet boundary conditions are mentioned briefly.
Keywords: exponential formulas, parabolic $p$-Laplacian, Nonlinear diffusion, homogeneous $p$-Laplacian, nonlinear semigroups.
Mathematics Subject Classification: Primary: 35K, 47H20; Secondary: 37L05, 47H0.
Citation: Matthew B. Rudd. Statistical exponential formulas for homogeneous diffusion. Communications on Pure & Applied Analysis, 2015, 14 (1) : 269-284. doi: 10.3934/cpaa.2015.14.269
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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.   Google Scholar

[5]

G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations,, \emph{Asymptotic Anal.}, 4 (1991), 271.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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E. DiBenedetto, Degenerate Parabolic Equations,, Universitext, (1993).  doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[11]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I,, \emph{J. Differential Geom.}, 33 (1991), 635.   Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

Y. Giga, Surface Evolution Equations, vol. 99 of Monographs in Mathematics,, Birkh\, (2006).   Google Scholar

[18]

J. A. Goldstein, Semigroups of Linear Operators and Applications,, Oxford Mathematical Monographs, (1985).   Google Scholar

[19]

A. Grigor'yan, Heat Kernel and Analysis on Manifolds, vol. 47 of AMS/IP Studies in Advanced Mathematics,, American Mathematical Society, (2009).   Google Scholar

[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.  Google Scholar

[21]

D. Hartenstine and M. Rudd, Statistical functional equations and $p$-harmonious functions,, \emph{Adv. Nonlinear Stud.}, 13 (2013), 191.   Google Scholar

[22]

J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations,, Oxford Mathematical Monographs, (1993).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

B. Kawohl, Variations on the $p$-Laplacian,, in \emph{Nonlinear elliptic partial differential equations}, (2011), 35.  doi: 10.1090/conm/540/10657.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[34]

G. F. Lawler, Random Walk and the Heat Equation, vol. 55 of Student Mathematical Library,, American Mathematical Society, (2010).   Google Scholar

[35]

P. D. Lax, Functional Analysis,, Pure and Applied Mathematics, (2002).   Google Scholar

[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.  Google Scholar

[37]

E. Le Gruyer and J. C. Archer, Harmonious extensions,, \emph{SIAM J. Math. Anal.}, 29 (1998), 279.  doi: 10.1137/S0036141095294067.  Google Scholar

[38]

G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co. Inc., (1996).  doi: 10.1142/3302.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[50]

W. P. Ziemer, Weakly Differentiable Functions,, vol. 120 of Graduate Texts in Mathematics, (1989).  doi: 10.1007/978-1-4612-1015-3.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[5]

G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations,, \emph{Asymptotic Anal.}, 4 (1991), 271.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[10]

E. DiBenedetto, Degenerate Parabolic Equations,, Universitext, (1993).  doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[11]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I,, \emph{J. Differential Geom.}, 33 (1991), 635.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

Y. Giga, Surface Evolution Equations, vol. 99 of Monographs in Mathematics,, Birkh\, (2006).   Google Scholar

[18]

J. A. Goldstein, Semigroups of Linear Operators and Applications,, Oxford Mathematical Monographs, (1985).   Google Scholar

[19]

A. Grigor'yan, Heat Kernel and Analysis on Manifolds, vol. 47 of AMS/IP Studies in Advanced Mathematics,, American Mathematical Society, (2009).   Google Scholar

[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.  Google Scholar

[21]

D. Hartenstine and M. Rudd, Statistical functional equations and $p$-harmonious functions,, \emph{Adv. Nonlinear Stud.}, 13 (2013), 191.   Google Scholar

[22]

J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations,, Oxford Mathematical Monographs, (1993).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

B. Kawohl, Variations on the $p$-Laplacian,, in \emph{Nonlinear elliptic partial differential equations}, (2011), 35.  doi: 10.1090/conm/540/10657.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[34]

G. F. Lawler, Random Walk and the Heat Equation, vol. 55 of Student Mathematical Library,, American Mathematical Society, (2010).   Google Scholar

[35]

P. D. Lax, Functional Analysis,, Pure and Applied Mathematics, (2002).   Google Scholar

[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.  Google Scholar

[37]

E. Le Gruyer and J. C. Archer, Harmonious extensions,, \emph{SIAM J. Math. Anal.}, 29 (1998), 279.  doi: 10.1137/S0036141095294067.  Google Scholar

[38]

G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co. Inc., (1996).  doi: 10.1142/3302.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[50]

W. P. Ziemer, Weakly Differentiable Functions,, vol. 120 of Graduate Texts in Mathematics, (1989).  doi: 10.1007/978-1-4612-1015-3.  Google Scholar

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