Abstract

We establish a new energy inequality for the difference of two semiconvex functions in a bounded open convex set D of RnR^n. This inequality is applied to the L2L^2-approximation problem of the gradient of the unknown solution of the nonlinear elliptic partial differential equation provided that the latter solution is a semiconvex function in D.

Author information

Contact details are reproduced from the original publication and may be historical.

Kakha Shashiashvili

I. Javakhishvili Tbilisi State University, Faculty of Exact and Natural Sciences, 2 University Street, Tbilisi 0186, Georgia

Malkhaz Shashiashvili

A. Razmadze Mathematical Institute, I. Javakhishvili Tbilisi State University, 2 University Street, Tbilisi 0186, Georgia

mshashiashvili@yahoo.com

Keywords

  • Semiconvex and semiconcave functions
  • energy inequality
  • L-2-approximation
  • gradient of semiconvex function
  • convex envelope

Mathematics Subject Classification

26B25, 35J60, 49L25

References

  1. O. Alvarez, J.-M. Lasry, P.-L. Lions: Convex viscosity solutions and state constraints, J. Math. Pures Appl. (9) 76(3) (1997) 265–288.https://doi.org/10.1016/S0021-7824(97)89952-7Google Scholar
  2. G. Barles, E. R. Jakobsen: Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations, SIAM J. Numer. Anal. 43(2) (2005) 540–558.https://doi.org/10.1137/S003614290343815XGoogle Scholar
  3. P. Cannarsa, C. Sinestrari: Semiconcave Functions, Hamilton–Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and their Applications 58, Birkhäuser, Boston (2004).https://doi.org/10.1007/b138356Google Scholar
  4. L. A. Caffarelli, P. E. Souganidis: A rate of convergence for monotone finite difference approximations to fully nonlinear, uniformly elliptic PDEs, Commun. Pure Appl. Math. 61(1) (2008) 1–17.https://doi.org/10.1002/cpa.20208Google Scholar
  5. L. C. Evans, R. F. Gariepy: Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton (1992).Google Scholar
  6. W. H. Fleming, H. M. Soner: Controlled Markov Processes and Viscosity Solutions, 2nd Ed., Stochastic Modelling and Applied Probability 25, Springer, New York (2006).Google Scholar
  7. D. Gilbarg, N. S. Trudinger: Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften 224, Springer, Berlin (1977).https://doi.org/10.1007/978-3-642-96379-7Google Scholar
  8. P. C. Hammer: Approximation of convex surfaces by algebraic surfaces, Mathematika, Lond. 10 (1963) 64–71.https://doi.org/10.1112/S0025579300003375Google Scholar
  9. S. Hussain, M. Shashiashvili: Discrete time hedging of the American option, Math. Finance 20(4) (2010) 647–670.https://doi.org/10.1111/j.1467-9965.2010.00415.xGoogle Scholar
  10. F. Kadhi, A. Trad: Characterization and approximation of the convex envelope of a function, J. Optimization Theory Appl. 110(2) (2001) 457–466.https://doi.org/10.1023/A:1017591716397Google Scholar
  11. A. U. Kennington: Power concavity and boundary value problems, Indiana Univ. Math. J. 34(3) (1985) 687–704.https://doi.org/10.1512/iumj.1985.34.34036Google Scholar
  12. N. J. Korevaar: Convex solutions to nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 32(4) (1983) 603–614.https://doi.org/10.1512/iumj.1983.32.32042Google Scholar
  13. N. V. Krylov: Controlled Diffusion Processes, Applications of Mathematics 14, Springer, Berlin (1980).https://doi.org/10.1007/978-1-4612-6051-6Google Scholar
  14. N. V. Krylov: The rate of convergence of finite-difference approximations for Bellman equations with Lipschitz coefficients, Appl. Math. Optim. 52(3) (2005) 365–399.https://doi.org/10.1007/s00245-005-0832-3Google Scholar
  15. P.-L. Lions: Optimal control of diffusion processes and Hamilton–Jacobi–Bellman equations. III: regularity of the optimal cost function, in: Nonlinear Partial Differential Equations and Their Applications (Paris, 1981–82), H. Brézis et al. (ed.), Res. Notes Math. 93, Pitman, Boston (1983) 95–205.Google Scholar
  16. P.-L. Lions: Optimal control of reflected diffusion processes: an example of state constraints, in: Stochastic differential systems (Bad Honnef, 1985), N. Christopeit et al. (ed.), Lecture Notes in Control and Inform. Sci. 78, Springer, Berlin (1986) 269–276.https://doi.org/10.1007/BFb0041170Google Scholar
  17. P.-L. Lions, N. S. Trudinger: Linear oblique derivative problems for the uniformly elliptic Hamilton–Jacobi–Bellman equation, Math. Z. 191(1) (1986) 1–15.https://doi.org/10.1007/BF01163605Google Scholar
  18. A. M. Oberman: The convex envelope is the solution of a nonlinear obstacle problem, Proc. Amer. Math. Soc. 135(6) (2007) 1689–1694.https://doi.org/10.1090/S0002-9939-07-08887-9Google Scholar
  19. A. M. Oberman: Computing the convex envelope using a nonlinear partial differential equation, Math. Models Methods Appl. Sci. 18(5) (2008) 759–780.https://doi.org/10.1142/S0218202508002851Google Scholar
  20. M. S. Saleem, M. Shahiashvili: Subsolutions that are close in the uniform norm are close in the Sobolev norm as well, Appl. Math. Optim. 66(1) (2012) 1–25.https://doi.org/10.1007/s00245-012-9161-5Google Scholar
  21. K. Shashiashvili, M. Shashiashvili: Estimation of the derivative of the convex function by means of its uniform approximation, JIPAM, J. Inequal. Pure Appl. Math. 6(4) (2005) Article 113, 10 pp.Google Scholar

Suggested citation

K. Shashiashvili, M. Shashiashvili. “From the Uniform Approximation of a Solution of the PDE to the L^2-Approximation of the Gradient of the Solution.” Journal of Convex Analysis 21 (2014), No. 1, 237–252. https://doi.org/10.68381/jca21013

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2014. Rights now held by Banach Press LLC.