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Uniqueness for Lp-viscosity solutions for uniformly parabolic Isaacs equations with measurable lower order terms
127 Vincent Hall, University of Minnesota, Minneapolis, MN, 55455 |
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.
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. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [6] |
S. Koike, Perron's method for Lp-viscosity solutions, Saitama Math. J., 23 (2005), 9-28. Google Scholar |
| [7] |
N. V. Krylov, Nonlinear Elliptic and Parabolic Equations of Second Order, Nauka, Moscow, 1985 in Russian; English translation: Reidel, Dordrecht, 1987. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [11] |
N. V. Krylov, To the theory of viscosity solutions for uniformly parabolic Isaacs equations, Methods and Applications of Analysis, 22 (2015), 259-280. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [14] | G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore, 1996. Google Scholar |
| [15] |
A. Świȩch, Wp1-interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations, 2 (1997), 1005-1027. Google Scholar |
| [16] |
N. S. Trudinger, Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations, Rev. Mat. Iberoamericana, 4 (1988), 453-468. Google Scholar |
| [17] |
L. Wang, On the regularity of fully nonlinear parabolic equations: Ⅱ, Comm. Pure Appl. Math., 45 (1992), 141-178. Google Scholar |
show all references
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. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [6] |
S. Koike, Perron's method for Lp-viscosity solutions, Saitama Math. J., 23 (2005), 9-28. Google Scholar |
| [7] |
N. V. Krylov, Nonlinear Elliptic and Parabolic Equations of Second Order, Nauka, Moscow, 1985 in Russian; English translation: Reidel, Dordrecht, 1987. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [11] |
N. V. Krylov, To the theory of viscosity solutions for uniformly parabolic Isaacs equations, Methods and Applications of Analysis, 22 (2015), 259-280. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [14] | G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore, 1996. Google Scholar |
| [15] |
A. Świȩch, Wp1-interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations, 2 (1997), 1005-1027. Google Scholar |
| [16] |
N. S. Trudinger, Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations, Rev. Mat. Iberoamericana, 4 (1988), 453-468. Google Scholar |
| [17] |
L. Wang, On the regularity of fully nonlinear parabolic equations: Ⅱ, Comm. Pure Appl. Math., 45 (1992), 141-178. Google Scholar |
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