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

Agnid Banerjee; Nicola Garofalo

doi:10.3934/cpaa.2015.14.1

Previous Article
Very weak solutions of singular porous medium equations with measure data
CPAA Home
This Issue
Next Article
Preface

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

Agnid Banerjee ^1, and Nicola Garofalo ^2,

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

Abstract
Related Papers

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 $.

Keywords: Dirichlet problem, large time behavior., Normalized $p$ laplacian evolution, Wiener criterion, tug of war games.

Mathematics Subject Classification: Primary: 35J25; Secondary: 35J7.

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

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors