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

show all references

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

[1]

Ángel Arroyo, Joonas Heino, Mikko Parviainen. Tug-of-war games with varying probabilities and the normalized p(x)-laplacian. Communications on Pure & Applied Analysis, 2017, 16 (3) : 915-944. doi: 10.3934/cpaa.2017044

[2]

Juan J. Manfredi, Julio D. Rossi, Stephanie J. Somersille. An obstacle problem for Tug-of-War games. Communications on Pure & Applied Analysis, 2015, 14 (1) : 217-228. doi: 10.3934/cpaa.2015.14.217

[3]

Ivana Gómez, Julio D. Rossi. Tug-of-war games and the infinity Laplacian with spatial dependence. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1959-1983. doi: 10.3934/cpaa.2013.12.1959

[4]

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

[5]

Junyong Eom, Ryuichi Sato. Large time behavior of ODE type solutions to parabolic $ p $-Laplacian type equations. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4373-4386. doi: 10.3934/cpaa.2020199

[6]

Fang Liu. The eigenvalue problem for a class of degenerate operators related to the normalized $ p $-Laplacian. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021155

[7]

Everaldo S. de Medeiros, Jianfu Yang. Asymptotic behavior of solutions to a perturbed p-Laplacian problem with Neumann condition. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 595-606. doi: 10.3934/dcds.2005.12.595

[8]

Fang Liu. An inhomogeneous evolution equation involving the normalized infinity Laplacian with a transport term. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2395-2421. doi: 10.3934/cpaa.2018114

[9]

Zhong Tan, Zheng-An Yao. The existence and asymptotic behavior of the evolution p-Laplacian equations with strong nonlinear sources. Communications on Pure & Applied Analysis, 2004, 3 (3) : 475-490. doi: 10.3934/cpaa.2004.3.475

[10]

Salvatore A. Marano, Nikolaos S. Papageorgiou. Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter. Communications on Pure & Applied Analysis, 2013, 12 (2) : 815-829. doi: 10.3934/cpaa.2013.12.815

[11]

Goro Akagi, Kei Matsuura. Well-posedness and large-time behaviors of solutions for a parabolic equation involving $p(x)$-Laplacian. Conference Publications, 2011, 2011 (Special) : 22-31. doi: 10.3934/proc.2011.2011.22

[12]

Raúl Ferreira, Julio D. Rossi. Decay estimates for a nonlocal $p-$Laplacian evolution problem with mixed boundary conditions. Discrete & Continuous Dynamical Systems, 2015, 35 (4) : 1469-1478. doi: 10.3934/dcds.2015.35.1469

[13]

Mostafa Ghelichi, A. M. Goltabar, H. R. Tavakoli, A. Karamodin. Neuro-fuzzy active control optimized by Tug of war optimization method for seismically excited benchmark highway bridge. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 333-351. doi: 10.3934/naco.2020029

[14]

Toyohiko Aiki, Adrian Muntean. Large time behavior of solutions to a moving-interface problem modeling concrete carbonation. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1117-1129. doi: 10.3934/cpaa.2010.9.1117

[15]

Zhenhua Guo, Wenchao Dong, Jinjing Liu. Large-time behavior of solution to an inflow problem on the half space for a class of compressible non-Newtonian fluids. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2133-2161. doi: 10.3934/cpaa.2019096

[16]

Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure & Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012

[17]

Francesca Colasuonno, Benedetta Noris. A p-Laplacian supercritical Neumann problem. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3025-3057. doi: 10.3934/dcds.2017130

[18]

Francesca Colasuonno, Fausto Ferrari. The Soap Bubble Theorem and a $ p $-Laplacian overdetermined problem. Communications on Pure & Applied Analysis, 2020, 19 (2) : 983-1000. doi: 10.3934/cpaa.2020045

[19]

Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075

[20]

Po-Chun Huang, Shin-Hwa Wang, Tzung-Shin Yeh. Classification of bifurcation diagrams of a $P$-Laplacian nonpositone problem. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2297-2318. doi: 10.3934/cpaa.2013.12.2297

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (398)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]