American Institute of Mathematical Sciences

November 2018, 17(6): 2495-2516. doi: 10.3934/cpaa.2018119

Uniqueness for Lp-viscosity solutions for uniformly parabolic Isaacs equations with measurable lower order terms

 127 Vincent Hall, University of Minnesota, Minneapolis, MN, 55455

Received  October 2017 Revised  February 2018 Published  June 2018

In this article we present several results concerning uniqueness of $C$-viscosity and $L_{p}$-viscosity solutions for fully nonlinear parabolic equations. In case of the Isaacs equations we allow lower order terms to have just measurable bounded coefficients. Higher-order coefficients are assumed to be Hölder continuous in $x$ with exponent slightly less than $1/2$. This case is treated by using stability of maximal and minimal $L_{p}$-viscosity solutions.

Citation: N. V. Krylov. Uniqueness for Lp-viscosity solutions for uniformly parabolic Isaacs equations with measurable lower order terms. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2495-2516. doi: 10.3934/cpaa.2018119
References:
 [1] 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. (N.S.), 27 (1992), 1-67. [2] M. G. Crandall, M. Kocan, P. L. Lions and A. Świȩch, Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations, Electron. J. Differential Equations, 24 (1999), 1-20. [3] M. G. Crandall, M. Kocan and A. Świȩch, Lp-theory for fully nonlinear uniformly parabolic equations, Comm. Partial Differential Equations, 25 (2000), 1997-2053. [4] Hongjie Dong, N. V. Krylov and Xu Li, On fully nonlinear elliptic and parabolic equations in domains with VMO coefficients, Algebra i Analiz, 24 (2012), 54-95, St. Petersburg Math. J., 24 (2013), 39-69. [5] R. Jensen and A. Świȩch, Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE, Comm. on Pure Appl. Analysis, 4 (2005), 199-207. [6] S. Koike, Perron's method for Lp-viscosity solutions, Saitama Math. J., 23 (2005), 9-28. [7] N. V. Krylov, Nonlinear Elliptic and Parabolic Equations of Second Order, Nauka, Moscow, 1985 in Russian; English translation: Reidel, Dordrecht, 1987. [8] N. V. Krylov, , Some Lp-estimates for elliptic and parabolic operators with measurable coefficients, Discrete and Continuous Dynamical Systems, Series B, 17 (2012), 2073-2090. [9] N. V Krylov, On the existence of Wp2 solutions for fully nonlinear elliptic equations under relaxed convexity assumptions, Comm. Partial Differential Equations, 38 (2013), 687-710. [10] N. V. Krylov, On C1+α regularity of solutions of Isaacs parabolic equations with VMO coefficients, Nonlinear Differential Equations and Applications, NoDEA, 21 (2014), 63-85. [11] N. V. Krylov, To the theory of viscosity solutions for uniformly parabolic Isaacs equations, Methods and Applications of Analysis, 22 (2015), 259-280. [12] N. V. Krylov, C1+α-regularity of viscosity solutions of general nonlinear parabolic equations, Problemy Matematicheskogo Analiza, 93, June 2018, 3-23; English translation in Journal of Mathematical Sciences, New York (Springer), 232 (2018), 403-427, http://arXiv.org/abs/1710.08884. [13] N. V. Krylov, On the existence of Wp1,2 solutions for fully nonlinear parabolic equations under either relaxed or no convexity assumptions, Harvard University, Center of Mathematical Sciences and Applications, Nonlinear Equation Publication, http://arXiv.org/abs/1705.02400. [14] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore, 1996. [15] A. Świȩch, Wp1-interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations, 2 (1997), 1005-1027. [16] N. S. Trudinger, Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations, Rev. Mat. Iberoamericana, 4 (1988), 453-468. [17] L. Wang, On the regularity of fully nonlinear parabolic equations: Ⅱ, Comm. Pure Appl. Math., 45 (1992), 141-178.

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References:
 [1] 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. (N.S.), 27 (1992), 1-67. [2] M. G. Crandall, M. Kocan, P. L. Lions and A. Świȩch, Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations, Electron. J. Differential Equations, 24 (1999), 1-20. [3] M. G. Crandall, M. Kocan and A. Świȩch, Lp-theory for fully nonlinear uniformly parabolic equations, Comm. Partial Differential Equations, 25 (2000), 1997-2053. [4] Hongjie Dong, N. V. Krylov and Xu Li, On fully nonlinear elliptic and parabolic equations in domains with VMO coefficients, Algebra i Analiz, 24 (2012), 54-95, St. Petersburg Math. J., 24 (2013), 39-69. [5] R. Jensen and A. Świȩch, Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE, Comm. on Pure Appl. Analysis, 4 (2005), 199-207. [6] S. Koike, Perron's method for Lp-viscosity solutions, Saitama Math. J., 23 (2005), 9-28. [7] N. V. Krylov, Nonlinear Elliptic and Parabolic Equations of Second Order, Nauka, Moscow, 1985 in Russian; English translation: Reidel, Dordrecht, 1987. [8] N. V. Krylov, , Some Lp-estimates for elliptic and parabolic operators with measurable coefficients, Discrete and Continuous Dynamical Systems, Series B, 17 (2012), 2073-2090. [9] N. V Krylov, On the existence of Wp2 solutions for fully nonlinear elliptic equations under relaxed convexity assumptions, Comm. Partial Differential Equations, 38 (2013), 687-710. [10] N. V. Krylov, On C1+α regularity of solutions of Isaacs parabolic equations with VMO coefficients, Nonlinear Differential Equations and Applications, NoDEA, 21 (2014), 63-85. [11] N. V. Krylov, To the theory of viscosity solutions for uniformly parabolic Isaacs equations, Methods and Applications of Analysis, 22 (2015), 259-280. [12] N. V. Krylov, C1+α-regularity of viscosity solutions of general nonlinear parabolic equations, Problemy Matematicheskogo Analiza, 93, June 2018, 3-23; English translation in Journal of Mathematical Sciences, New York (Springer), 232 (2018), 403-427, http://arXiv.org/abs/1710.08884. [13] N. V. Krylov, On the existence of Wp1,2 solutions for fully nonlinear parabolic equations under either relaxed or no convexity assumptions, Harvard University, Center of Mathematical Sciences and Applications, Nonlinear Equation Publication, http://arXiv.org/abs/1705.02400. [14] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore, 1996. [15] A. Świȩch, Wp1-interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations, 2 (1997), 1005-1027. [16] N. S. Trudinger, Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations, Rev. Mat. Iberoamericana, 4 (1988), 453-468. [17] L. Wang, On the regularity of fully nonlinear parabolic equations: Ⅱ, Comm. Pure Appl. Math., 45 (1992), 141-178.
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