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An evolution equation involving the normalized $P$-Laplacian

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  • This paper considers an initial-boundary value problem for the evolution equation associated with the normalized $p$-Laplacian. There exists a unique viscosity solution $u,$ which is globally Lipschitz continuous with respect to $t$ and locally with respect to $x.$ Moreover, we study the long time behavior of the viscosity solution $u$ and compute numerical solutions of the problem.
    Mathematics Subject Classification: Primary: 35K92, 35K61, 35K65, 35Q91; Secondary: 65M06.

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  • [1]

    A. Almansa, F. Cao, Y. Gousseau and B. Rougé, Interpolation of Digital Elevation Models Using AMLE and Related Methods, IEEE Transaction on Geoscience and Remote Sensing, 40 (2002), 314-325.

    [2]

    G. Barles, Nonlinear Neumann boundary conditions for quasilinear degenerate elliptic equations and applications, J. Differential Equations, 154 (1999), 191-224.

    [3]

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

    [4]

    I. Birindelli and F. Demengel, First eigenvalue and maximum principle for fully nonlinear singular operators, Adv. Differential Equations, 11 (2006), 91-119.

    [5]

    V. Caselles, J. M. Morel and C. Sbert, An axiomatic approach to image interpolation, IEEE Trans. Image Process., 7 (1998), 376-386.doi: doi:10.1109/83.661188.

    [6]

    Y. G. Chen and E. DiBenedetto, On the local behavior of solutions of singular parabolic equations, Arch. Rational Mech. Anal., 103 (1988), 319-345.

    [7]

    Y. G. Chen, Y. Giga and S. Goto, Remarks on viscosity solutions for evolution equations, Proc. Japan Acad. Ser. A Math. Sci., 67 (1991), 323-328.doi: doi:10.3792/pjaa.67.323.

    [8]

    L. Collatz, "The Numerical Treatment of Differential Equations," Springer-Verlag, Berlin-Göttingen-Heidelberg, 1966.

    [9]

    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.

    [10]

    E. DiBenedetto, "Degenerate Parabolic Equations," Springer-Verlag, New York, 1993.doi: doi:10.1515/crll.1985.357.1.

    [11]

    E. DiBenedetto and M. A. Herrero, Nonnegative solutions of the evolution $p$-Laplacian equation. Initial traces and Cauchy problem when $1, Arch. Rational Mech. Anal., 111 (1990), 225-290.

    [12]

    E. DiBenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math., 357 (1985), 1-22.

    [13]

    K. Does, "An Evolution Equation Involving the Normalized $p$-Laplacian," Ph.D thesis, Universität zu Köln, 2009.

    [14]

    P. Dupius and H. Ishii, On oblique derivative problems for fully nonlinear second-order elliptic partial differential equations on nonsmooth domains, Nonlinear Anal., 12 (1990), 1123-1138.

    [15]

    L. C. Evans, The 1-Laplacian, the $\infty$-Laplacian and differential games, Contemp. Math., 446 (2007), 245-254.

    [16]

    L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," CRC Press, Boca Raton, Ann Arbor and London, 1992.

    [17]

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

    [18]

    Y. Giga, "Surface Evolution Equation - a Level Set Method," Birkhäuser, Basel, 2006.

    [19]

    C. Grossmann and H.-G. Roos, "Numerik Partieller Differentialgleichungen," Teubner Verlag, Wiesbaden, 1994.

    [20]

    W. Hackbusch, "Theorie und Numerik Elliptischer Differentialgleichungen," Teubner Verlag, Wiesbaden, 1996.

    [21]

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

    [22]

    B. Kawohl, Variational versus PDE-based approaches in mathematical image processing, CRM Proceedings and Lecture Notes, 44 (2006), 113-126.

    [23]

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

    [24]

    O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva, "Linear and Quasilinear Equations of Parabolic Type," American Mathematical Society, Providence, Rhode Island, 1968.

    [25]

    T. Leonori and J. M. UrbanoGrowth Conditions and Uniqueness of the Cauchy Problem for the Evolutionary Infinity Laplacian, preprint, arXiv:0809.2523.

    [26]

    G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co. Pte. Ltd, Singapore, 1996.

    [27]

    J. J. Manfredi, M. Parviainen and J. D. RossiAn asymptotic mean value characterization for a class of nonlinear parabolic equations related to tug-of-war games, available at: http://math.tkk.fi/ mjparvia/index.html

    [28]

    A. M. Oberman, A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions, Math. Comp., 74 (2005), 1217-1230.

    [29]

    M. Ohnuma and K. Sato, Singular degenerate parabolic equations with applications to the $p$-Laplace diffusion equation, Comm. Partial Differential Equations, 22 (1997), 381-411.

    [30]

    Y. Peres, O. Schramm, S. Sheffield and D. B. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc., 22 (2009), 167-210.doi: doi:10.1090/S0894-0347-08-00606-1.

    [31]

    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: doi:10.1215/00127094-2008-048.

    [32]

    W. Rudin, "Principles of Mathematical Analysis," McGraw-Hill Book Company, New York, San Fransisco, 1964.

    [33]

    X. Xu, On the Cauchy problem for a singular parabolic equation, Pacific J. Math., 174 (1996), 277-294.

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