# American Institute of Mathematical Sciences

January  2015, 14(1): 1-21. doi: 10.3934/cpaa.2015.14.1

## On the Dirichlet boundary value problem for the normalized $p$-laplacian evolution

 1 Department of Mathematics, Purdue University, West Lafayette, IN 47907, United States 2 Dipartimento di Ingegneria Civile, Edile e Ambientale (DICEA), Università di Padova, 35131 Padova, Italy

Received  April 2014 Revised  June 2014 Published  September 2014

In this paper, we study the potential theoretic aspects of the normalized $p$-Laplacian evolution, see (1.1) below. A systematic study of such equation was recently started in [1], [4] and [25]. Via the classical Perron approach, we address the question of solvability of the Cauchy-Dirichlet problem with "very weak" assumptions on the boundary of the domain. The regular boundary points for the Dirichlet problem are characterized in terms of barriers. For $p \geq 2$, in the case of space - time cylinder $G \times (0,T)$, we show that $(x,t) \in \partial G \times (0, T]$ is a regular boundary point if and only if $x \in \partial G$ is a a regular boundary point for the p-Laplacian. This latter operator is the steady state corresponding to the evolution (1.1) below. Consequently, when $p\geq 2$ the Cauchy- Dirichlet problem for (1.1) can be solved in cylinders whose section is regular for the $p$-Laplacian. This can be thought of as an analogue of the results obtained in [17] for the standard parabolic $p$-Laplacian div$(|Du|^{p-2}Du) - u_t = 0$.
Citation: Agnid Banerjee, Nicola Garofalo. On the Dirichlet boundary value problem for the normalized $p$-laplacian evolution. Communications on Pure & Applied Analysis, 2015, 14 (1) : 1-21. doi: 10.3934/cpaa.2015.14.1
##### References:
 [1] A. Banerjee and N. Garofalo, Gradient bounds and monotonicity of the energy for some nonlinear singular diffusion equations, Indiana Univ. Math. J., 62 (2013), 699-736. doi: 10.1512/iumj.2013.62.4969.  Google Scholar [2] Y.-G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Diff. Geom., 33 (1991), 749-786.  Google Scholar [3] M. Crandall, M. Kocan and A. Swiech, $L^p$-theory for fully nonlinear uniformly parabolic equations, Comm. Partial Differential Equations, 25 (2000), 1997-2053. doi: 10.1080/03605300008821576.  Google Scholar [4] K. Does, An evolution equation involving the normalized $p$-Laplacian, Comm. Pure Appl. Anal., 10 (2011), 361-396. doi: 10.3934/cpaa.2011.10.361.  Google Scholar [5] L. Evans and R. Gariepy, Wiener's criterion for the heat equation, Arch. Rational Mech. Anal., 78 (1982), 293-314. doi: 10.1007/BF00249583.  Google Scholar [6] L. C. Evans and J. Spruck, Motions of level sets by mean curvature, Part I, J. Diff. Geom., 33 (1991), 635-681.  Google Scholar [7] E. Fabes, N. Garofalo and E. Lanconelli, Wiener's criterion for divergence form parabolic operators with $C^1$-Dini continuous coefficients, Duke Math. J., 59 (1989), 191-232. doi: 10.1215/S0012-7094-89-05906-1.  Google Scholar [8] A. Friedman, Parabolic equations of the second order, Trans. Amer. Math. Soc., 93 (1959), 509-530.  Google Scholar [9] R. Gariepy and W. P. Ziemer, A regularity condition at the boundary for solutions of quasilinear elliptic equations, Arch. Rat. Mech. Anal., 67 (1977), 25-39.  Google Scholar [10] S. Granlund, P. Lindqvist and O. Martio, Note on the PWB-method in the nonlinear case, Pacific J. Math., 125 (1986), 381-395.  Google Scholar [11] M. Gruber, Harnack inequalities for solutions of general second order parabolic equations and estimates of their Hölder constants, Math. Z., 185 (1984), 23-43. doi: 10.1007/BF01214972.  Google Scholar [12] J. Heinonen, T. Kilpelainen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993.  Google Scholar [13] C. Imbert and L. Silvestre, Introduction to fully nonlinear parabolic equations, in An Introduction to the Kähler-Ricci Flow, Lecture Notes in Mathematics, 2086 (2013), 7-88. doi: 10.1007/978-3-319-00819-6_2.  Google Scholar [14] P. Juutinen, Decay estimates in sup norm for the solutions to a nonlinear evolution equation, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 144 (2014), 557-566. doi: 10.1017/S0308210512001163.  Google Scholar [15] 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.  Google Scholar [16] P. Juutinen, P. Lindqvist and J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasilinear equation, SIAM J. Math. Anal., 33 (2001), 699-717. doi: 10.1137/S0036141000372179.  Google Scholar [17] T. Kilpelainen and P. Lindqvist, On the Dirichlet boundary value problem for a degenerate parabolic equation, SIAM J. Math. Anal., 27 (1996), 661-683. doi: 10.1137/0527036.  Google Scholar [18] T. Kilpelainen and J. Maly, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161. doi: 10.1007/BF02392793.  Google Scholar [19] N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), 161-175 (Russian).  Google Scholar [20] O. Ladyzhenskaja and N. Uraltseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis Academic Press, New York-London, 1968.  Google Scholar [21] O. Ladyzhenskaja, V. A. Solonnikov and N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1967. Google Scholar [22] G. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.  Google Scholar [23] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219. doi: 10.1016/0362-546X(88)90053-3.  Google Scholar [24] J. Maly and W. P. Ziemer, Fine Regularity of Solutions of Elliptic Partial Differential Equations, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 51 (1997), xiv+291 pp. ISBN: 0-8218-0335-2. doi: 10.1090/surv/051.  Google Scholar [25] 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.  Google Scholar [26] V. G. Mazya, The continuity at a boundary point of the solutions of quasi-linear elliptic equations, Vestnik Leningrad. Univ., 25 (1970), 42-55.  Google Scholar [27] K. Miller, Barriers on cones for uniformly elliptic operators, Ann. Mat. Pura Appl., 76 (1967), 93-105.  Google Scholar [28] M. Ohnuma and K. Sato, Singular degenerate parabolic equations with applications, Comm. Partial Differential Equations, 22 (1997), 381-441. doi: 10.1080/03605309708821268.  Google Scholar [29] O. Perron, Die Stabilittsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728. doi: 10.1007/BF01194662.  Google Scholar [30] J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math., 111 (1964), 247-302.  Google Scholar [31] I. Skrypnik, On the Wiener criterion for quasilinear degenerate parabolic equations, Dokl. Akad. Nauk, 398 (2004), 458-461.  Google Scholar [32] W. Sternberg, Über die Gleichung der Wärmeleitung, Math. Ann., 101 (1929), 394-398. doi: 10.1007/BF01454850.  Google Scholar [33] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150. doi: 10.1016/0022-0396(84)90105-0.  Google Scholar [34] G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains, J. Funct. Anal., 59 (1984), 572-611. doi: 10.1016/0022-1236(84)90066-1.  Google Scholar [35] L. Wang, On the regularity theory of fully nonlinear parabolic equations: I, Comm. Pure Appl. Math., 45 (1992), 27-76. doi: 10.1002/cpa.3160450103.  Google Scholar

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##### References:
 [1] A. Banerjee and N. Garofalo, Gradient bounds and monotonicity of the energy for some nonlinear singular diffusion equations, Indiana Univ. Math. J., 62 (2013), 699-736. doi: 10.1512/iumj.2013.62.4969.  Google Scholar [2] Y.-G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Diff. Geom., 33 (1991), 749-786.  Google Scholar [3] M. Crandall, M. Kocan and A. Swiech, $L^p$-theory for fully nonlinear uniformly parabolic equations, Comm. Partial Differential Equations, 25 (2000), 1997-2053. doi: 10.1080/03605300008821576.  Google Scholar [4] K. Does, An evolution equation involving the normalized $p$-Laplacian, Comm. Pure Appl. Anal., 10 (2011), 361-396. doi: 10.3934/cpaa.2011.10.361.  Google Scholar [5] L. Evans and R. Gariepy, Wiener's criterion for the heat equation, Arch. Rational Mech. Anal., 78 (1982), 293-314. doi: 10.1007/BF00249583.  Google Scholar [6] L. C. Evans and J. Spruck, Motions of level sets by mean curvature, Part I, J. Diff. Geom., 33 (1991), 635-681.  Google Scholar [7] E. Fabes, N. Garofalo and E. Lanconelli, Wiener's criterion for divergence form parabolic operators with $C^1$-Dini continuous coefficients, Duke Math. J., 59 (1989), 191-232. doi: 10.1215/S0012-7094-89-05906-1.  Google Scholar [8] A. Friedman, Parabolic equations of the second order, Trans. Amer. Math. Soc., 93 (1959), 509-530.  Google Scholar [9] R. Gariepy and W. P. Ziemer, A regularity condition at the boundary for solutions of quasilinear elliptic equations, Arch. Rat. Mech. Anal., 67 (1977), 25-39.  Google Scholar [10] S. Granlund, P. Lindqvist and O. Martio, Note on the PWB-method in the nonlinear case, Pacific J. Math., 125 (1986), 381-395.  Google Scholar [11] M. Gruber, Harnack inequalities for solutions of general second order parabolic equations and estimates of their Hölder constants, Math. Z., 185 (1984), 23-43. doi: 10.1007/BF01214972.  Google Scholar [12] J. Heinonen, T. Kilpelainen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993.  Google Scholar [13] C. Imbert and L. Silvestre, Introduction to fully nonlinear parabolic equations, in An Introduction to the Kähler-Ricci Flow, Lecture Notes in Mathematics, 2086 (2013), 7-88. doi: 10.1007/978-3-319-00819-6_2.  Google Scholar [14] P. Juutinen, Decay estimates in sup norm for the solutions to a nonlinear evolution equation, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 144 (2014), 557-566. doi: 10.1017/S0308210512001163.  Google Scholar [15] 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.  Google Scholar [16] P. Juutinen, P. Lindqvist and J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasilinear equation, SIAM J. Math. Anal., 33 (2001), 699-717. doi: 10.1137/S0036141000372179.  Google Scholar [17] T. Kilpelainen and P. Lindqvist, On the Dirichlet boundary value problem for a degenerate parabolic equation, SIAM J. Math. Anal., 27 (1996), 661-683. doi: 10.1137/0527036.  Google Scholar [18] T. Kilpelainen and J. Maly, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161. doi: 10.1007/BF02392793.  Google Scholar [19] N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), 161-175 (Russian).  Google Scholar [20] O. Ladyzhenskaja and N. Uraltseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis Academic Press, New York-London, 1968.  Google Scholar [21] O. Ladyzhenskaja, V. A. Solonnikov and N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1967. Google Scholar [22] G. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.  Google Scholar [23] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219. doi: 10.1016/0362-546X(88)90053-3.  Google Scholar [24] J. Maly and W. P. Ziemer, Fine Regularity of Solutions of Elliptic Partial Differential Equations, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 51 (1997), xiv+291 pp. ISBN: 0-8218-0335-2. doi: 10.1090/surv/051.  Google Scholar [25] 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.  Google Scholar [26] V. G. Mazya, The continuity at a boundary point of the solutions of quasi-linear elliptic equations, Vestnik Leningrad. Univ., 25 (1970), 42-55.  Google Scholar [27] K. Miller, Barriers on cones for uniformly elliptic operators, Ann. Mat. Pura Appl., 76 (1967), 93-105.  Google Scholar [28] M. Ohnuma and K. Sato, Singular degenerate parabolic equations with applications, Comm. Partial Differential Equations, 22 (1997), 381-441. doi: 10.1080/03605309708821268.  Google Scholar [29] O. Perron, Die Stabilittsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728. doi: 10.1007/BF01194662.  Google Scholar [30] J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math., 111 (1964), 247-302.  Google Scholar [31] I. Skrypnik, On the Wiener criterion for quasilinear degenerate parabolic equations, Dokl. Akad. Nauk, 398 (2004), 458-461.  Google Scholar [32] W. Sternberg, Über die Gleichung der Wärmeleitung, Math. Ann., 101 (1929), 394-398. doi: 10.1007/BF01454850.  Google Scholar [33] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150. doi: 10.1016/0022-0396(84)90105-0.  Google Scholar [34] G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains, J. Funct. Anal., 59 (1984), 572-611. doi: 10.1016/0022-1236(84)90066-1.  Google Scholar [35] L. Wang, On the regularity theory of fully nonlinear parabolic equations: I, Comm. Pure Appl. Math., 45 (1992), 27-76. doi: 10.1002/cpa.3160450103.  Google Scholar
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