# American Institute of Mathematical Sciences

January  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, Bull. Amer. Math. Soc., 41 (2004), 439-505. doi: 10.1090/S0273-0979-04-01035-3.  Google Scholar [2] M. Bardi and F. Da Lio, On the strong maximum principle for fully nonlinear degenerate elliptic equations, Arch. Math., 73 (1999), 276-285. doi: 10.1007/s000130050399.  Google Scholar [3] G. Barles and J. Busca, Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term, Comm. Partial Differential Equations, 26 (2001), 2323-2337. doi: 10.1081/PDE-100107824.  Google Scholar [4] G. Barles, E. Rouy and P. E. Souganidis, Remarks on the Dirichlet problem for quasilinear elliptic and parabolic equations, Stochastic Analysis, Control, Optimization and Applications, Birkhäuser, Boston, (1999), 209-222.  Google Scholar [5] M. G. Crandall, A visit with the $\infty$-Laplace equation, Lecture Notes in Math., 1927, Springer, Berlin, (2008), 75-122. doi: 10.1007/978-3-540-75914-0_3.  Google Scholar [6] M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 277 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar [7] M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42. doi: 10.2307/1999343.  Google Scholar [8] Y.-G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geometry, 33 (1991), 749-786.  Google Scholar [9] H. Ishii and S. Koike, Viscosity solutions of a system of nonlinear second-order elliptic PDEs arising in switching games, Funkcial. Ekvac., 34 (1991), 143-155.  Google Scholar [10] R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rational Mech. Math., 101 (1988), 1-27. doi: 10.1007/BF00281780.  Google Scholar [11] R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Rational Mech. Anal., 123 (1993), 51-74. doi: 10.1007/BF00386368.  Google Scholar [12] P. Juutinen, P. Lindqvist and J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasilinear eqation, SIAM J. Math. Anal., 33 (2001), 699-717. doi: 10.1137/S0036141000372179.  Google Scholar [13] B. Kawohl and N. Kutev, Comparison principle and Lipschitz regularity for viscosity solutions of nonlinear partial differential equations, Funkcial. Ekvac., 43 (2000), 241-253.  Google Scholar [14] B. Kawohl and N. Kutev, Comparison principle for viscosity solutions of fully nonlinear, degenerate elliptic equations, Comm. Partial Differential Equations, 32 (2007), 1209-1224. doi: 10.1080/03605300601113043.  Google Scholar [15] S. Koike, A Beginner's Guide to the Theory of Viscosity Solutions, MSJ memoir 13, Math. Soc. Japan, 2004.  Google Scholar [16] P. Lindqvist, Notes on the $p$-Laplace equation, Report. University of Jyväskylä, Department of Mathematics and Statics, 102, 2006.  Google Scholar [17] Y. Luo and A. Eberhard, Comparison principle for viscosity solutions of elliptic equations via fuzzy sum rule, J. Math. Anal. Appl., 307 (2005), 736-752. doi: 10.1016/j.jmaa.2005.01.055.  Google Scholar [18] R. T. Rockafellar, Convex Analysis, Princeton Math. Series, 28, Princeton University Press, 1970.  Google Scholar [19] N. S. Trudinger, Comparison principles and pointwise estimates for viscosity solutions, Rev. Mat. Iberoamericana, 4 (1988), 453-468. doi: 10.4171/RMI/80.  Google Scholar

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##### References:
 [1] G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc., 41 (2004), 439-505. doi: 10.1090/S0273-0979-04-01035-3.  Google Scholar [2] M. Bardi and F. Da Lio, On the strong maximum principle for fully nonlinear degenerate elliptic equations, Arch. Math., 73 (1999), 276-285. doi: 10.1007/s000130050399.  Google Scholar [3] G. Barles and J. Busca, Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term, Comm. Partial Differential Equations, 26 (2001), 2323-2337. doi: 10.1081/PDE-100107824.  Google Scholar [4] G. Barles, E. Rouy and P. E. Souganidis, Remarks on the Dirichlet problem for quasilinear elliptic and parabolic equations, Stochastic Analysis, Control, Optimization and Applications, Birkhäuser, Boston, (1999), 209-222.  Google Scholar [5] M. G. Crandall, A visit with the $\infty$-Laplace equation, Lecture Notes in Math., 1927, Springer, Berlin, (2008), 75-122. doi: 10.1007/978-3-540-75914-0_3.  Google Scholar [6] M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 277 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar [7] M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42. doi: 10.2307/1999343.  Google Scholar [8] Y.-G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geometry, 33 (1991), 749-786.  Google Scholar [9] H. Ishii and S. Koike, Viscosity solutions of a system of nonlinear second-order elliptic PDEs arising in switching games, Funkcial. Ekvac., 34 (1991), 143-155.  Google Scholar [10] R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rational Mech. Math., 101 (1988), 1-27. doi: 10.1007/BF00281780.  Google Scholar [11] R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Rational Mech. Anal., 123 (1993), 51-74. doi: 10.1007/BF00386368.  Google Scholar [12] P. Juutinen, P. Lindqvist and J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasilinear eqation, SIAM J. Math. Anal., 33 (2001), 699-717. doi: 10.1137/S0036141000372179.  Google Scholar [13] B. Kawohl and N. Kutev, Comparison principle and Lipschitz regularity for viscosity solutions of nonlinear partial differential equations, Funkcial. Ekvac., 43 (2000), 241-253.  Google Scholar [14] B. Kawohl and N. Kutev, Comparison principle for viscosity solutions of fully nonlinear, degenerate elliptic equations, Comm. Partial Differential Equations, 32 (2007), 1209-1224. doi: 10.1080/03605300601113043.  Google Scholar [15] S. Koike, A Beginner's Guide to the Theory of Viscosity Solutions, MSJ memoir 13, Math. Soc. Japan, 2004.  Google Scholar [16] P. Lindqvist, Notes on the $p$-Laplace equation, Report. University of Jyväskylä, Department of Mathematics and Statics, 102, 2006.  Google Scholar [17] Y. Luo and A. Eberhard, Comparison principle for viscosity solutions of elliptic equations via fuzzy sum rule, J. Math. Anal. Appl., 307 (2005), 736-752. doi: 10.1016/j.jmaa.2005.01.055.  Google Scholar [18] R. T. Rockafellar, Convex Analysis, Princeton Math. Series, 28, Princeton University Press, 1970.  Google Scholar [19] N. S. Trudinger, Comparison principles and pointwise estimates for viscosity solutions, Rev. Mat. Iberoamericana, 4 (1988), 453-468. doi: 10.4171/RMI/80.  Google Scholar
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