Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The minmax principle and $ W^{2,p}$ regularity for solutions of the simplest Isaacs equations

Author: Jay Kovats
Journal: Proc. Amer. Math. Soc. 140 (2012), 2803-2815
MSC (2010): Primary 35B65, 35J60, 49N60, 49N70
Published electronically: February 21, 2012
MathSciNet review: 2910767
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we consider the simplest uniformly elliptic Isaacs equations and prove that when the control matrix is appropriately separable, $ C^{2}$ solutions satisfy an interior $ W^{2,p}$ estimate for all $ 0<p<\infty $.

References [Enhancements On Off] (What's this?)

  • [C] L. Caffarelli, Interior a Priori Estimates for Viscosity Solutions of Fully Nonlinear Equations, Annals of Math. (2) 130 (1989), 189-213. MR 1005611 (90i:35046)
  • [CC1] L. Caffarelli and X. Cabre, Fully Nonlinear Elliptic Equations, Amer. Math. Soc., Providence, RI, 1995. MR 1351007 (96h:35046)
  • [CC2] -, Interior $ C^{2,\alpha }$ Regularity for a Class of Nonconvex Fully Nonlinear Elliptic Equations, J. Math. Pures Appl. 82 (2003), 573-612. MR 1995493 (2004f:35049)
  • [CY] L. Caffarelli and Y. Yuan, A Priori Estimates for Solutions of Fully Nonlinear Equations with Convex Level Set, Indiana University Math. Jour. 49 (2) (2000), 681-695. MR 1793687 (2002b:35049)
  • [GT] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, New York, 1983. MR 737190 (86c:35035)
  • [I] H. Ishii, On Uniqueness and Existence of Viscosity Solutions of Fully Nonlinear Second-Order Elliptic PDEs, Comm. Pure Appl. Math. 42 (1989), 14-45. MR 973743 (89m:35070)
  • [K] J. Kovats, Value Functions and the Dirichlet Problem for Isaacs Equation in a Smooth Domain, Transactions of the AMS 361 (2009), 4045-4076. MR 2500878 (2010c:49057)
  • [Kr] N.V. Krylov, Boundedly Nonhomogeneous Elliptic and Parabolic Equations, vol. 20, Izv. Acad. Nauk. SSSR Ser. Mat. (1983), pp. 459-492, English transl. in Math. USSR Izv. MR 688919 (85g:35046)
  • [L] F.-H. Lin, Second Derivative $ L^{p}$ Estimates for Elliptic Equations of Nondivergent Type, Proceedings of the Amer. Math. Soc. 96 (3) (1984), pp. 447-451. MR 822437 (88b:35058)
  • [NV1] N. Nadirashvili and S. Vlăduţ, Nonclassical Solutions of Fully Nonlinear Elliptic Equations, Geometric and Functional Analysis 17 (2007), pp. 1283-1296. MR 2373018 (2008m:35121)
  • [NV2] -, Singular Solutions to Fully Nonlinear Elliptic Equations, Jour. Math. Pures Appl. 89 (2008), pp. 107-113. MR 2391642 (2009a:35080)
  • [NV3] -, Singular Solutions of Hessian Fully Nonlinear Elliptic Equations, Adv. Math. 228 (2011), pp. 1718-1741.
  • [N] M. Nisio, Stochastic Differential Games and Viscosity Solutions of Isaacs Equations, Nagoya Math. J. 110 (1988), pp. 163-184. MR 945913 (90b:93100)
  • [R] R.T. Rockafellar, Convex Analysis, Princeton University Press, 1970. MR 0274683 (43:445)
  • [T] N.S. Trudinger, On the Twice Differentiability of Viscosity Solutions of Nonlinear Elliptic Equations, Bull. Austral. Math. Soc. 39 (1989), pp. 443-447. MR 995142 (90f:35038)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35B65, 35J60, 49N60, 49N70

Retrieve articles in all journals with MSC (2010): 35B65, 35J60, 49N60, 49N70

Additional Information

Jay Kovats
Affiliation: Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, Florida 32901

Received by editor(s): March 14, 2011
Published electronically: February 21, 2012
Communicated by: James E. Colliander
Article copyright: © Copyright 2012 American Mathematical Society

American Mathematical Society