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Statistical exponential formulas for homogeneous diffusion

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  • 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.
    Mathematics Subject Classification: Primary: 35K, 47H20; Secondary: 37L05, 47H05.


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  • [1]

    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.doi: 10.1007/s00208-008-0297-1.


    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.doi: 10.1007/s00526-007-0117-6.


    L. Alvarez, F. Guichard, P.-L. Lions and J.-M. Morel, Axioms and fundamental equations of image processing, Arch. Rational Mech. Anal., 123 (1993), 199-257.doi: 10.1007/BF00375127.


    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, Berlin, 1997, Lectures given at the 2nd C.I.M.E. Session held in Montecatini Terme, June 12-20, 1995, Edited by I. Capuzzo Dolcetta and P. L. Lions, Fondazione C.I.M.E.. [C.I.M.E. Foundation].


    G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., 4 (1991), 271-283.


    G. Barles and C. Georgelin, A simple proof of convergence for an approximation scheme for computing motions by mean curvature, SIAM J. Numer. Anal., 32 (1995), 484-500.doi: 10.1137/0732020.


    F. Catté, F. Dibos and G. Koepfler, A morphological scheme for mean curvature motion and applications to anisotropic diffusion and motion of level sets, SIAM J. Numer. Anal., 32 (1995), 1895-1909.doi: 10.1137/0732085.


    M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), 265-298.


    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. (N.S.), 27 (1992), 1-67.doi: 10.1090/S0273-0979-1992-00266-5.


    E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993.doi: 10.1007/978-1-4612-0895-2.


    L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom., 33 (1991), 635-681.


    L. C. Evans and J. Spruck, Motion of level sets by mean curvature. II, Trans. Amer. Math. Soc., 330 (1992), 321-332.doi: 10.2307/2154167.


    L. C. Evans and J. Spruck, Motion of level sets by mean curvature. III, J. Geom. Anal., 2 (1992), 121-150.doi: 10.1007/BF02921385.


    L. C. Evans, Convergence of an algorithm for mean curvature motion, Indiana Univ. Math. J., 42 (1993), 533-557.doi: 10.1512/iumj.1993.42.42024.


    L. C. Evans and J. Spruck, Motion of level sets by mean curvature. IV, J. Geom. Anal., 5 (1995), 77-114.doi: 10.1007/BF02926443.


    Y. Giga, S. Goto, H. Ishii and M.-H. Sato, Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains, Indiana Univ. Math. J., 40 (1991), 443-470.doi: 10.1512/iumj.1991.40.40023.


    Y. Giga, Surface Evolution Equations, vol. 99 of Monographs in Mathematics, Birkhäuser Verlag, Basel, 2006, A level set approach.


    J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1985.


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


    D. Hartenstine and M. Rudd, Asymptotic statistical characterizations of $p$-harmonic functions of two variables, Rocky Mountain J. Math., 41 (2011), 493-504.doi: 10.1216/RMJ-2011-41-2-493.


    D. Hartenstine and M. Rudd, Statistical functional equations and $p$-harmonious functions, Adv. Nonlinear Stud., 13 (2013), 191-207.


    J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1993, Oxford Science Publications.


    T. Ilmanen, P. Sternberg and W. P. Ziemer, Equilibrium solutions to generalized motion by mean curvature, J. Geom. Anal., 8 (1998), 845-858. Dedicated to the memory of Fred Almgren.doi: 10.1007/BF02922673.


    H. Ishii, G. E. Pires and P. E. Souganidis, Threshold dynamics type approximation schemes for propagating fronts, J. Math. Soc. Japan, 51 (1999), 267-308.doi: 10.2969/jmsj/05120267.


    V. Julin and P. Juutinen, A new proof for the equivalence of weak and viscosity solutions for the $p$-Laplace equation, Comm. Partial Differential Equations, 37 (2012), 934-946.doi: 10.1080/03605302.2011.615878.


    P. Juutinen, $p$-harmonic approximation of functions of least gradient, Indiana Univ. Math. J., 54 (2005), 1015-1030.doi: 10.1512/iumj.2005.54.2658.


    P. Juutinen and B. Kawohl, On the evolution governed by the infinity Laplacian, Math. Ann., 335 (2006), 819-851.doi: 10.1007/s00208-006-0766-3.


    P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation, SIAM J. Math. Anal., 33 (2001), 699-717.doi: 10.1137/S0036141000372179.


    B. Kawohl and N. Kutev, Comparison principle for viscosity solutions of fully nonlinear, degenerate elliptic equations, Comm. Partial Differential Equations, 32 (2007), 1209-1224.doi: 10.1080/03605300601113043.


    B. Kawohl, Variations on the $p$-Laplacian, in Nonlinear elliptic partial differential equations, vol. 540 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2011, 35-46.doi: 10.1090/conm/540/10657.


    B. Kawohl, J. Manfredi and M. Parviainen, Solutions of nonlinear PDEs in the sense of averages, J. Math. Pures Appl., 97 (2012), 173-188.doi: 10.1016/j.matpur.2011.07.001.


    B. Kawohl and N. Kutev, Comparison principle and Lipschitz regularity for viscosity solutions of some classes of nonlinear partial differential equations, Funkcial. Ekvac., 43 (2000), 241-253.


    R. V. Kohn and S. Serfaty, A deterministic-control-based approach to motion by curvature, Comm. Pure Appl. Math., 59 (2006), 344-407.doi: 10.1002/cpa.20101.


    G. F. Lawler, Random Walk and the Heat Equation, vol. 55 of Student Mathematical Library, American Mathematical Society, Providence, RI, 2010.


    P. D. Lax, Functional Analysis, Pure and Applied Mathematics, Wiley-Interscience, John Wiley & Sons, New York, 2002.


    E. Le Gruyer, On absolutely minimizing Lipschitz extensions and PDE $\Delta_\infty(u)=0$, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 29-55.doi: 10.1007/s00030-006-4030-z.


    E. Le Gruyer and J. C. Archer, Harmonious extensions, SIAM J. Math. Anal., 29 (1998), 279-292.doi: 10.1137/S0036141095294067.


    G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. Inc., River Edge, NJ, 1996.doi: 10.1142/3302.


    P. Lindqvist, Notes on the $p$-Laplace equation, vol. 102 of Report, University of Jyväskylä Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, 2006.


    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, SIAM J. Math. Anal., 42 (2010), 2058-2081.doi: 10.1137/100782073.


    B. Merriman, J. K. Bence and S. J. Osher, Motion of multiple functions: a level set approach, J. Comput. Phys., 112 (1994), 334-363.doi: 10.1006/jcph.1994.1105.


    A. M. Oberman, A convergent monotone difference scheme for motion of level sets by mean curvature, Numer. Math., 99 (2004), 365-379.doi: 10.1007/s00211-004-0566-1.


    A. M. Oberman, A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions, Math. Comp., 74 (2005), 1217-1230.doi: 10.1090/S0025-5718-04-01688-6.


    S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49.doi: 10.1016/0021-9991(88)90002-2.


    A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983.doi: 10.1007/978-1-4612-5561-1.


    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.doi: 10.1215/00127094-2008-048.


    S. J. Ruuth and B. Merriman, Convolution-generated motion and generalized Huygens' principles for interface motion, SIAM J. Appl. Math., 60 (2000), 868-890.doi: 10.1137/S003613999833397X.


    P. Sternberg and W. P. Ziemer, Generalized motion by curvature with a Dirichlet condition, J. Differential Equations, 114 (1994), 580-600.doi: 10.1006/jdeq.1994.1162.


    N. T. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, vol. 100 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1992.


    W. P. Ziemer, Weakly Differentiable Functions, vol. 120 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1989.doi: 10.1007/978-1-4612-1015-3.

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