2015, 14(1): 133-142. doi: 10.3934/cpaa.2015.14.133

Remarks on the comparison principle for quasilinear PDE with no zeroth order terms

1. 

Mathematical Institute, Tohoku University, Aoba, Sendai, 980-8578, Japan, Japan

Received  January 2014 Revised  April 2014 Published  September 2014

A comparison principle for viscosity solutions of second-order quasilinear elliptic partial di erential equations with no zeroth order terms is shown. A di erent transformation from that of Barles and Busca in [3] is adapted to enable us to deal with slightly more general equations.
Citation: Shigeaki Koike, Takahiro Kosugi. Remarks on the comparison principle for quasilinear PDE with no zeroth order terms. Communications on Pure & Applied Analysis, 2015, 14 (1) : 133-142. doi: 10.3934/cpaa.2015.14.133
References:
[1]

G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions,, \emph{Bull. Amer. Math. Soc.}, 41 (2004), 439. doi: 10.1090/S0273-0979-04-01035-3.

[2]

M. Bardi and F. Da Lio, On the strong maximum principle for fully nonlinear degenerate elliptic equations,, \emph{Arch. Math.}, 73 (1999), 276. doi: 10.1007/s000130050399.

[3]

G. Barles and J. Busca, Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term,, \emph{Comm. Partial Differential Equations}, 26 (2001), 2323. doi: 10.1081/PDE-100107824.

[4]

G. Barles, E. Rouy and P. E. Souganidis, Remarks on the Dirichlet problem for quasilinear elliptic and parabolic equations,, \emph{Stochastic Analysis, (1999), 209.

[5]

M. G. Crandall, A visit with the $\infty$-Laplace equation,, \emph{Lecture Notes in Math.}, 1927 (2008), 75. doi: 10.1007/978-3-540-75914-0_3.

[6]

M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations,, \emph{Bull. Amer. Math. Soc.}, 277 (1992), 1. doi: 10.1090/S0273-0979-1992-00266-5.

[7]

M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations,, \emph{Trans. Amer. Math. Soc.}, 277 (1983), 1. doi: 10.2307/1999343.

[8]

Y.-G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations,, \emph{J. Differential Geometry}, 33 (1991), 749.

[9]

H. Ishii and S. Koike, Viscosity solutions of a system of nonlinear second-order elliptic PDEs arising in switching games,, \emph{Funkcial. Ekvac.}, 34 (1991), 143.

[10]

R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations,, \emph{Arch. Rational Mech. Math.}, 101 (1988), 1. doi: 10.1007/BF00281780.

[11]

R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient,, \emph{Arch. Rational Mech. Anal.}, 123 (1993), 51. doi: 10.1007/BF00386368.

[12]

P. Juutinen, P. Lindqvist and J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasilinear eqation,, \emph{SIAM J. Math. Anal.}, 33 (2001), 699. doi: 10.1137/S0036141000372179.

[13]

B. Kawohl and N. Kutev, Comparison principle and Lipschitz regularity for viscosity solutions of nonlinear partial differential equations,, \emph{Funkcial. Ekvac.}, 43 (2000), 241.

[14]

B. Kawohl and N. Kutev, Comparison principle for viscosity solutions of fully nonlinear, degenerate elliptic equations,, \emph{Comm. Partial Differential Equations}, 32 (2007), 1209. doi: 10.1080/03605300601113043.

[15]

S. Koike, A Beginner's Guide to the Theory of Viscosity Solutions,, MSJ memoir \textbf{13}, 13 (2004).

[16]

P. Lindqvist, Notes on the $p$-Laplace equation, Report., \emph{University of Jyv\, 102 (2006).

[17]

Y. Luo and A. Eberhard, Comparison principle for viscosity solutions of elliptic equations via fuzzy sum rule,, \emph{J. Math. Anal. Appl.}, 307 (2005), 736. doi: 10.1016/j.jmaa.2005.01.055.

[18]

R. T. Rockafellar, Convex Analysis,, Princeton Math. Series, 28 (1970).

[19]

N. S. Trudinger, Comparison principles and pointwise estimates for viscosity solutions,, \emph{Rev. Mat. Iberoamericana}, 4 (1988), 453. doi: 10.4171/RMI/80.

show all references

References:
[1]

G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions,, \emph{Bull. Amer. Math. Soc.}, 41 (2004), 439. doi: 10.1090/S0273-0979-04-01035-3.

[2]

M. Bardi and F. Da Lio, On the strong maximum principle for fully nonlinear degenerate elliptic equations,, \emph{Arch. Math.}, 73 (1999), 276. doi: 10.1007/s000130050399.

[3]

G. Barles and J. Busca, Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term,, \emph{Comm. Partial Differential Equations}, 26 (2001), 2323. doi: 10.1081/PDE-100107824.

[4]

G. Barles, E. Rouy and P. E. Souganidis, Remarks on the Dirichlet problem for quasilinear elliptic and parabolic equations,, \emph{Stochastic Analysis, (1999), 209.

[5]

M. G. Crandall, A visit with the $\infty$-Laplace equation,, \emph{Lecture Notes in Math.}, 1927 (2008), 75. doi: 10.1007/978-3-540-75914-0_3.

[6]

M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations,, \emph{Bull. Amer. Math. Soc.}, 277 (1992), 1. doi: 10.1090/S0273-0979-1992-00266-5.

[7]

M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations,, \emph{Trans. Amer. Math. Soc.}, 277 (1983), 1. doi: 10.2307/1999343.

[8]

Y.-G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations,, \emph{J. Differential Geometry}, 33 (1991), 749.

[9]

H. Ishii and S. Koike, Viscosity solutions of a system of nonlinear second-order elliptic PDEs arising in switching games,, \emph{Funkcial. Ekvac.}, 34 (1991), 143.

[10]

R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations,, \emph{Arch. Rational Mech. Math.}, 101 (1988), 1. doi: 10.1007/BF00281780.

[11]

R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient,, \emph{Arch. Rational Mech. Anal.}, 123 (1993), 51. doi: 10.1007/BF00386368.

[12]

P. Juutinen, P. Lindqvist and J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasilinear eqation,, \emph{SIAM J. Math. Anal.}, 33 (2001), 699. doi: 10.1137/S0036141000372179.

[13]

B. Kawohl and N. Kutev, Comparison principle and Lipschitz regularity for viscosity solutions of nonlinear partial differential equations,, \emph{Funkcial. Ekvac.}, 43 (2000), 241.

[14]

B. Kawohl and N. Kutev, Comparison principle for viscosity solutions of fully nonlinear, degenerate elliptic equations,, \emph{Comm. Partial Differential Equations}, 32 (2007), 1209. doi: 10.1080/03605300601113043.

[15]

S. Koike, A Beginner's Guide to the Theory of Viscosity Solutions,, MSJ memoir \textbf{13}, 13 (2004).

[16]

P. Lindqvist, Notes on the $p$-Laplace equation, Report., \emph{University of Jyv\, 102 (2006).

[17]

Y. Luo and A. Eberhard, Comparison principle for viscosity solutions of elliptic equations via fuzzy sum rule,, \emph{J. Math. Anal. Appl.}, 307 (2005), 736. doi: 10.1016/j.jmaa.2005.01.055.

[18]

R. T. Rockafellar, Convex Analysis,, Princeton Math. Series, 28 (1970).

[19]

N. S. Trudinger, Comparison principles and pointwise estimates for viscosity solutions,, \emph{Rev. Mat. Iberoamericana}, 4 (1988), 453. doi: 10.4171/RMI/80.

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