January  2011, 10(1): 361-396. doi: 10.3934/cpaa.2011.10.361

An evolution equation involving the normalized $P$-Laplacian

1. 

Mathematisches Institut, Universität zu Köln, 50923 Köln, Germany

Received  February 2010 Revised  May 2010 Published  November 2010

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.
Citation: Kerstin Does. An evolution equation involving the normalized $P$-Laplacian. Communications on Pure & Applied Analysis, 2011, 10 (1) : 361-396. doi: 10.3934/cpaa.2011.10.361
References:
[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. Google Scholar

[2]

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

[3]

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

[4]

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

[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.  Google Scholar

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

[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.  Google Scholar

[8]

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

[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. Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[13]

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

[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. Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[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.  Google Scholar

[22]

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

[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.  Google Scholar

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

[25]

T. Leonori and J. M. Urbano, Growth Conditions and Uniqueness of the Cauchy Problem for the Evolutionary Infinity Laplacian,, preprint, ().   Google Scholar

[26]

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

[27]

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,, available at: http://math.tkk.fi/ mjparvia/index.html, ().   Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[33]

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

show all references

References:
[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. Google Scholar

[2]

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

[3]

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

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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. Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[13]

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

[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. Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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. Google Scholar

[25]

T. Leonori and J. M. Urbano, Growth Conditions and Uniqueness of the Cauchy Problem for the Evolutionary Infinity Laplacian,, preprint, ().   Google Scholar

[26]

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

[27]

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,, available at: http://math.tkk.fi/ mjparvia/index.html, ().   Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[33]

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

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