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,, \emph{Indiana Univ. Math. J.}, 62 (2013), 699. 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,, \emph{J. Diff. Geom.}, 33 (1991), 749. Google Scholar [3] M. Crandall, M. Kocan and A. Swiech, $L^p$-theory for fully nonlinear uniformly parabolic equations,, \emph{Comm. Partial Differential Equations}, 25 (2000), 1997. doi: 10.1080/03605300008821576. Google Scholar [4] K. Does, An evolution equation involving the normalized $p$-Laplacian,, \emph{Comm. Pure Appl. Anal.}, 10 (2011), 361. doi: 10.3934/cpaa.2011.10.361. Google Scholar [5] L. Evans and R. Gariepy, Wiener's criterion for the heat equation,, \emph{Arch. Rational Mech. Anal.}, 78 (1982), 293. doi: 10.1007/BF00249583. Google Scholar [6] L. C. Evans and J. Spruck, Motions of level sets by mean curvature, Part I,, \emph{J. Diff. Geom.}, 33 (1991), 635. Google Scholar [7] E. Fabes, N. Garofalo and E. Lanconelli, Wiener's criterion for divergence form parabolic operators with $C^1$-Dini continuous coefficients,, \emph{Duke Math. J.}, 59 (1989), 191. doi: 10.1215/S0012-7094-89-05906-1. Google Scholar [8] A. Friedman, Parabolic equations of the second order,, \emph{Trans. Amer. Math. Soc.}, 93 (1959), 509. Google Scholar [9] R. Gariepy and W. P. Ziemer, A regularity condition at the boundary for solutions of quasilinear elliptic equations,, \emph{Arch. Rat. Mech. Anal.}, 67 (1977), 25. Google Scholar [10] S. Granlund, P. Lindqvist and O. Martio, Note on the PWB-method in the nonlinear case,, \emph{Pacific J. Math.}, 125 (1986), 381. Google Scholar [11] M. Gruber, Harnack inequalities for solutions of general second order parabolic equations and estimates of their Hölder constants,, \emph{Math. Z.}, 185 (1984), 23. doi: 10.1007/BF01214972. Google Scholar [12] J. Heinonen, T. Kilpelainen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations,, Oxford Science Publications, (1993). Google Scholar [13] C. Imbert and L. Silvestre, Introduction to fully nonlinear parabolic equations,, in \emph{An Introduction to the K\, 2086 (2013), 7. 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,, \emph{Proceedings of the Royal Society of Edinburgh: Section A Mathematics}, 144 (2014), 557. doi: 10.1017/S0308210512001163. Google Scholar [15] P. Juutinen and B. Kawohl, On the evolution governed by the infinity Laplacian,, \emph{Math. Ann.}, 335 (2006), 819. 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,, \emph{SIAM J. Math. Anal.}, 33 (2001), 699. doi: 10.1137/S0036141000372179. Google Scholar [17] T. Kilpelainen and P. Lindqvist, On the Dirichlet boundary value problem for a degenerate parabolic equation,, \emph{SIAM J. Math. Anal.}, 27 (1996), 661. doi: 10.1137/0527036. Google Scholar [18] T. Kilpelainen and J. Maly, The Wiener test and potential estimates for quasilinear elliptic equations,, \emph{Acta Math.}, 172 (1994), 137. 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,, \emph{Izv. Akad. Nauk SSSR Ser. Mat.}, 44 (1980), 161. Google Scholar [20] O. Ladyzhenskaja and N. Uraltseva, Linear and Quasilinear Elliptic Equations,, Translated from the Russian by Scripta Technica, (1968). Google Scholar [21] O. Ladyzhenskaja, V. A. Solonnikov and N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type,, Translations of Mathematical Monographs, (1967). Google Scholar [22] G. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co., (1996). doi: 10.1142/3302. Google Scholar [23] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations,, \emph{Nonlinear Anal.}, 12 (1988), 1203. 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, 51 (1997), 0. 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,, \emph{SIAM J. Math Anal.}, 42 (2010), 2058. doi: 10.1137/100782073. Google Scholar [26] V. G. Mazya, The continuity at a boundary point of the solutions of quasi-linear elliptic equations,, \emph{Vestnik Leningrad. Univ.}, 25 (1970), 42. Google Scholar [27] K. Miller, Barriers on cones for uniformly elliptic operators,, \emph{Ann. Mat. Pura Appl.}, 76 (1967), 93. Google Scholar [28] M. Ohnuma and K. Sato, Singular degenerate parabolic equations with applications,, \emph{Comm. Partial Differential Equations}, 22 (1997), 381. doi: 10.1080/03605309708821268. Google Scholar [29] O. Perron, Die Stabilittsfrage bei Differentialgleichungen,, \emph{Math. Z.}, 32 (1930), 703. doi: 10.1007/BF01194662. Google Scholar [30] J. Serrin, Local behavior of solutions of quasi-linear equations,, \emph{Acta Math.}, 111 (1964), 247. Google Scholar [31] I. Skrypnik, On the Wiener criterion for quasilinear degenerate parabolic equations,, \emph{Dokl. Akad. Nauk}, 398 (2004), 458. Google Scholar [32] W. Sternberg, Über die Gleichung der Wärmeleitung,, \emph{Math. Ann.}, 101 (1929), 394. doi: 10.1007/BF01454850. Google Scholar [33] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations,, \emph{J. Differential Equations}, 51 (1984), 126. 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,, \emph{J. Funct. Anal., 59 (1984), 572. doi: 10.1016/0022-1236(84)90066-1. Google Scholar [35] L. Wang, On the regularity theory of fully nonlinear parabolic equations: I,, \emph{Comm. Pure Appl. Math.}, 45 (1992), 27. 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,, \emph{Indiana Univ. Math. J.}, 62 (2013), 699. 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,, \emph{J. Diff. Geom.}, 33 (1991), 749. Google Scholar [3] M. Crandall, M. Kocan and A. Swiech, $L^p$-theory for fully nonlinear uniformly parabolic equations,, \emph{Comm. Partial Differential Equations}, 25 (2000), 1997. doi: 10.1080/03605300008821576. Google Scholar [4] K. Does, An evolution equation involving the normalized $p$-Laplacian,, \emph{Comm. Pure Appl. Anal.}, 10 (2011), 361. doi: 10.3934/cpaa.2011.10.361. Google Scholar [5] L. Evans and R. Gariepy, Wiener's criterion for the heat equation,, \emph{Arch. Rational Mech. Anal.}, 78 (1982), 293. doi: 10.1007/BF00249583. Google Scholar [6] L. C. Evans and J. Spruck, Motions of level sets by mean curvature, Part I,, \emph{J. Diff. Geom.}, 33 (1991), 635. Google Scholar [7] E. Fabes, N. Garofalo and E. Lanconelli, Wiener's criterion for divergence form parabolic operators with $C^1$-Dini continuous coefficients,, \emph{Duke Math. J.}, 59 (1989), 191. doi: 10.1215/S0012-7094-89-05906-1. Google Scholar [8] A. Friedman, Parabolic equations of the second order,, \emph{Trans. Amer. Math. Soc.}, 93 (1959), 509. Google Scholar [9] R. Gariepy and W. P. Ziemer, A regularity condition at the boundary for solutions of quasilinear elliptic equations,, \emph{Arch. Rat. Mech. Anal.}, 67 (1977), 25. Google Scholar [10] S. Granlund, P. Lindqvist and O. Martio, Note on the PWB-method in the nonlinear case,, \emph{Pacific J. Math.}, 125 (1986), 381. Google Scholar [11] M. Gruber, Harnack inequalities for solutions of general second order parabolic equations and estimates of their Hölder constants,, \emph{Math. Z.}, 185 (1984), 23. doi: 10.1007/BF01214972. Google Scholar [12] J. Heinonen, T. Kilpelainen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations,, Oxford Science Publications, (1993). Google Scholar [13] C. Imbert and L. Silvestre, Introduction to fully nonlinear parabolic equations,, in \emph{An Introduction to the K\, 2086 (2013), 7. 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,, \emph{Proceedings of the Royal Society of Edinburgh: Section A Mathematics}, 144 (2014), 557. doi: 10.1017/S0308210512001163. Google Scholar [15] P. Juutinen and B. Kawohl, On the evolution governed by the infinity Laplacian,, \emph{Math. Ann.}, 335 (2006), 819. 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,, \emph{SIAM J. Math. Anal.}, 33 (2001), 699. doi: 10.1137/S0036141000372179. Google Scholar [17] T. Kilpelainen and P. Lindqvist, On the Dirichlet boundary value problem for a degenerate parabolic equation,, \emph{SIAM J. Math. Anal.}, 27 (1996), 661. doi: 10.1137/0527036. Google Scholar [18] T. Kilpelainen and J. Maly, The Wiener test and potential estimates for quasilinear elliptic equations,, \emph{Acta Math.}, 172 (1994), 137. 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,, \emph{Izv. Akad. Nauk SSSR Ser. Mat.}, 44 (1980), 161. Google Scholar [20] O. Ladyzhenskaja and N. Uraltseva, Linear and Quasilinear Elliptic Equations,, Translated from the Russian by Scripta Technica, (1968). Google Scholar [21] O. Ladyzhenskaja, V. A. Solonnikov and N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type,, Translations of Mathematical Monographs, (1967). Google Scholar [22] G. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co., (1996). doi: 10.1142/3302. Google Scholar [23] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations,, \emph{Nonlinear Anal.}, 12 (1988), 1203. 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, 51 (1997), 0. 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,, \emph{SIAM J. Math Anal.}, 42 (2010), 2058. doi: 10.1137/100782073. Google Scholar [26] V. G. Mazya, The continuity at a boundary point of the solutions of quasi-linear elliptic equations,, \emph{Vestnik Leningrad. Univ.}, 25 (1970), 42. Google Scholar [27] K. Miller, Barriers on cones for uniformly elliptic operators,, \emph{Ann. Mat. Pura Appl.}, 76 (1967), 93. Google Scholar [28] M. Ohnuma and K. Sato, Singular degenerate parabolic equations with applications,, \emph{Comm. Partial Differential Equations}, 22 (1997), 381. doi: 10.1080/03605309708821268. Google Scholar [29] O. Perron, Die Stabilittsfrage bei Differentialgleichungen,, \emph{Math. Z.}, 32 (1930), 703. doi: 10.1007/BF01194662. Google Scholar [30] J. Serrin, Local behavior of solutions of quasi-linear equations,, \emph{Acta Math.}, 111 (1964), 247. Google Scholar [31] I. Skrypnik, On the Wiener criterion for quasilinear degenerate parabolic equations,, \emph{Dokl. Akad. Nauk}, 398 (2004), 458. Google Scholar [32] W. Sternberg, Über die Gleichung der Wärmeleitung,, \emph{Math. Ann.}, 101 (1929), 394. doi: 10.1007/BF01454850. Google Scholar [33] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations,, \emph{J. Differential Equations}, 51 (1984), 126. 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,, \emph{J. Funct. Anal., 59 (1984), 572. doi: 10.1016/0022-1236(84)90066-1. Google Scholar [35] L. Wang, On the regularity theory of fully nonlinear parabolic equations: I,, \emph{Comm. Pure Appl. Math.}, 45 (1992), 27. doi: 10.1002/cpa.3160450103. Google Scholar
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