Communications on Pure and Applied Analysis (CPAA)

An evolution equation involving the normalized $P$-Laplacian
Pages: 361 - 396, Volume 10, Issue 1, January 2011

doi:10.3934/cpaa.2011.10.361      Abstract        References        Full text (1438.7K)           Related Articles

Kerstin Does - Mathematisches Institut, Universität zu Köln, 50923 Köln, Germany (email)

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.       
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.       
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.
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.       
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.       
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. Urbano, Growth 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. 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
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.       
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.       
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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