Skip to main content
SpringerLink
Log in

Semiconvexity of Viscosity Solutions to Fully Nonlinear Evolution Equations via Discrete Games

  • Chapter
  • First Online:
Geometric Properties for Parabolic and Elliptic PDE's

Part of the book series: Springer INdAM Series ((SINDAMS,volume 47))

  • 860 Accesses

Abstract

In this paper, by using a discrete game interpretation of fully nonlinear parabolic equations proposed by Kohn and Serfaty (Commun Pure Appl Math 63(10):1298–1350, 2010), we show that the spatial semiconvexity of viscosity solutions is preserved for a class of fully nonlinear evolution equations with concave parabolic operators. We also reduce the game-theoretic argument to the viscous and inviscid Hamilton-Jacobi equations, categorizing the semiconvexity regularity of solutions in terms of semiconcavity of the Hamiltonian.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Alvarez, O., Lasry, J.-M., Lions, P.-L.: Convex viscosity solutions and state constraints. J. Math. Pures Appl. (9) 76(3), 265–288 (1997)

    Google Scholar 

  2. Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Systems & Control: Foundations & Applications. Birkhäuser, Boston (1997). With appendices by Maurizio Falcone and Pierpaolo Soravia

    Google Scholar 

  3. Barles, G., Biton, S., Bourgoing, M., Ley, O.: Uniqueness results for quasilinear parabolic equations through viscosity solutions’ methods. Calc. Var. Partial Differ. Equ. 18(2), 159–179 (2003)

    Article  MathSciNet  Google Scholar 

  4. Bian, B., Guan, P.: A microscopic convexity principle for nonlinear partial differential equations. Invent. Math. 177(2), 307–335 (2009)

    Article  MathSciNet  Google Scholar 

  5. Cannarsa, P., Sinestrari, C.: Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control. Progress in Nonlinear Differential Equations and Their Applications, vol. 58. Birkhäuser, Boston (2004)

    Google Scholar 

  6. Cheridito, P., Soner, H.M., Touzi, N., Victoir, N.: Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs. Commun. Pure Appl. Math. 60(7), 1081–1110 (2007)

    Article  MathSciNet  Google Scholar 

  7. Crandall, M.G., Newcomb, R., Tomita, Y.: Existence and uniqueness viscosity solutions of degenerate quasilinear elliptic equations in R n. Technical Summary Report, Wisconsin Univ-Madison Center for Mathematical Sciences (1988)

    MATH  Google Scholar 

  8. Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)

    Google Scholar 

  9. Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1998)

    Google Scholar 

  10. Evans, L.C., Spruck, J.: Motion of level sets by mean curvature. I. J. Differ. Geom. 33(3), 635–681 (1991)

    MathSciNet  MATH  Google Scholar 

  11. Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Stochastic Modelling and Applied Probability, vol. 25, 2nd edn. Springer, New York (2006)

    Google Scholar 

  12. Fujita, Y.: Hessian estimates for viscous Hamilton-Jacobi equations with the Ornstein-Uhlenbeck operator. Differ. Integr. Equ. 18(12), 1383–1396 (2005)

    MathSciNet  MATH  Google Scholar 

  13. Fujita, Y., Loreti, P.: Long-time behavior of solutions to Hamilton-Jacobi equations with quadratic gradient term. Nonlinear Differ. Equ. Appl. 16(6), 771–791 (2009)

    Article  MathSciNet  Google Scholar 

  14. Giga, Y.: Surface Evolution Equations. A Level Set Approach. Monographs in Mathematics, vol. 99. Birkhäuser, Basel (2006).

    Google Scholar 

  15. Giga, Y., Goto, S., Ishii, H., Sato, M.-H.: Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains. Indiana Univ. Math. J. 40(2), 443–470 (1991)

    Article  MathSciNet  Google Scholar 

  16. Imbert, C.: Convexity of solutions and C 1, 1 estimates for fully nonlinear elliptic equations. J. Math. Pures Appl. (9) 85(6), 791–807 (2006)

    Google Scholar 

  17. Ishige, K., Salani, P.: Is quasi-concavity preserved by heat flow? Arch. Math. 90, 450–460 (2008)

    Article  MathSciNet  Google Scholar 

  18. Ishige, K., Salani, P.: Convexity breaking of the free boundary for porous medium equations. Interfaces Free Bound. 12, 75–84 (2010)

    Article  MathSciNet  Google Scholar 

  19. Ishige, K., Nakagawa, K., Salani, P.: Power concavity in weakly coupled elliptic and parabolic systems. Nonlinear Anal. 131, 81–97 (2016)

    Article  MathSciNet  Google Scholar 

  20. Ishige, K., Liu, Q., Salani, P.: Parabolic Minkowski convolutions and concavity properties of viscosity solutions to fully nonlinear equations. J. Math. Pures Appl. (9) 141, 342–370 (2020)

    Google Scholar 

  21. Ishige, K., Nakagawa, K., Salani, P.: Spatial concavity of solutions to parabolic systems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 20(1), 291–313 (2020)

    MathSciNet  MATH  Google Scholar 

  22. Ishii, H.: Uniqueness of unbounded viscosity solution of Hamilton-Jacobi equations. Indiana Univ. Math. J. 33(5), 721–748 (1984)

    Article  MathSciNet  Google Scholar 

  23. Juutinen, P.: Concavity maximum principle for viscosity solutions of singular equations. Nonlinear Differ. Equ. Appl. 17(5), 601–618 (2010)

    Article  MathSciNet  Google Scholar 

  24. Kawohl, B.: Rearrangements and Convexity of Level Sets in PDE. Lecture Notes in Mathematics, vol. 1150. Springer, Berlin (1985)

    Google Scholar 

  25. Kawohl, B.: Qualitative properties of solutions to semilinear heat equations. Expo. Math. 4(3), 257–270 (1986)

    MathSciNet  MATH  Google Scholar 

  26. Kennington, A.U.: Power concavity and boundary value problems. Indiana Univ. Math. J. 34(3), 687–704 (1985)

    Article  MathSciNet  Google Scholar 

  27. Kohn, R.V., Serfaty, S.: A deterministic-control-based approach to motion by curvature. Commun. Pure Appl. Math. 59(3), 344–407 (2006)

    Article  MathSciNet  Google Scholar 

  28. Kohn, R.V., Serfaty, S.: A deterministic-control-based approach to fully nonlinear parabolic and elliptic equations. Commun. Pure Appl. Math. 63(10), 1298–1350 (2010)

    Article  MathSciNet  Google Scholar 

  29. Korevaar, N.J.: Convex solutions to nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J. 32(4), 603–614 (1983)

    Article  MathSciNet  Google Scholar 

  30. Kružkov, S.N.: Generalized solutions of nonlinear equations of the first order with several independent variables. II. Mat. Sb. (N.S.) 72(114), 108–134 (1967)

    Google Scholar 

  31. Kružkov, S.N.: Generalized solutions of Hamilton-Jacobi equations of eikonal type. I. Statement of the problems; existence, uniqueness and stability theorems; certain properties of the solutions. Mat. Sb. (N.S.) 98(140)(3(11)), 450–493, 496 (1975)

    Google Scholar 

  32. Lions, P.-L.: Generalized Solutions of Hamilton-Jacobi Equations. Research Notes in Mathematics, vol. 69. Pitman (Advanced Publishing Program), Boston (1982)

    Google Scholar 

  33. Liu, Q., Schikorra, A., Zhou, X.: A game-theoretic proof of convexity preserving properties for motion by curvature. Indiana Univ. Math. J. 65, 171–197 (2016)

    Article  MathSciNet  Google Scholar 

  34. Manfredi, J.J., Parviainen, M., Rossi, J.D.: An asymptotic mean value characterization for a class of nonlinear parabolic equations related to tug-of-war games. SIAM J. Math. Anal. 42(5), 2058–2081 (2010)

    Article  MathSciNet  Google Scholar 

  35. Manfredi, J.J., Parviainen, M., Rossi, J.D.: An asymptotic mean value characterization for p-harmonic functions. Proc. Am. Math. Soc. 138(3), 881–889 (2010)

    Article  MathSciNet  Google Scholar 

  36. Peres, Y., Sheffield, S.: Tug-of-war with noise: a game-theoretic view of the p-Laplacian. Duke Math. J. 145(1), 91–120 (2008)

    Article  MathSciNet  Google Scholar 

  37. Peres, Y., Schramm, O., Sheffield, S., Wilson, D.B.: Tug-of-war and the infinity Laplacian. J. Am. Math. Soc. 22(1), 167–210 (2009)

    Article  MathSciNet  Google Scholar 

  38. Sakaguchi, S.: Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14(3), 403–421 (1987/1988)

    Google Scholar 

  39. Strömberg, T.: Semiconcavity estimates for viscous Hamilton-Jacobi equations. Arch. Math. (Basel) 94(6), 579–589 (2010)

    Google Scholar 

  40. Triebel, H.: Theory of Function Spaces. II. Monographs in Mathematics, vol. 84. Birkhäuser, Basel (1992)

    Google Scholar 

  41. Villani, C.: Optimal Transport. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338. Springer, Berlin (2009). Old and new

    Google Scholar 

Download references

Acknowledgements

The author would like to thank the anonymous referees for careful reading and helpful comments. Several results related to this work were presented at the 6th Italian-Japanese Workshop on Geometric Properties for Parabolic and Elliptic PDEs held in Cortona, May 20–24, 2019. The author is grateful to the organizers for their invitation and hospitality.

The work of the author is supported by Japan Society for the Promotion of Science (JSPS) through the grants No. 16K17635 and No. 19K03574 and by Central Research Institute of Fukuoka University through the grant No. 177102.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Fukuoka University, Fukuoka, Japan

    Qing Liu

Authors
  1. Qing Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qing Liu .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli Federico II, Napoli, Italy

    Vincenzo Ferone

  2. Applied Mathematics and Informatics Course, Faculty of Advanced Science and Technology, Ryukoku University, Otsu, Shiga, Japan

    Tatsuki Kawakami

  3. DIMAI Dipartimento di Matematica e Informatica “U. Dini”, Università degli Studi di Firenze, Firenze, Italy

    Paolo Salani

  4. Department of Mathematics, Osaka City University, Osaka, Japan

    Futoshi Takahashi

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Liu, Q. (2021). Semiconvexity of Viscosity Solutions to Fully Nonlinear Evolution Equations via Discrete Games. In: Ferone, V., Kawakami, T., Salani, P., Takahashi, F. (eds) Geometric Properties for Parabolic and Elliptic PDE's. Springer INdAM Series, vol 47. Springer, Cham. https://doi.org/10.1007/978-3-030-73363-6_10

Download citation

Publish with us

Policies and ethics

Search

Navigation