Skip to main content
SpringerLink
Log in

Semiconcavity estimates for viscous Hamilton–Jacobi equations

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

We present sharp Hessian estimates of the form \({D^2 S^\varepsilon(t,x)\leq g(t)I}\) for the solution of the viscous Hamilton–Jacobi equation

$$\begin{array}{ll}S^\varepsilon_t+\frac{1}{2}|DS^\varepsilon|^2+V(x)-\varepsilon\Delta S^\varepsilon = 0\quad {\rm in} \, Q_T=(0,T]\times\, {\mathbb {R}^n}, \\ \qquad \qquad \qquad \qquad \quad \, S^\varepsilon(0,x) = S_0(x)\quad{\rm in}\, {\mathbb {R}^n}.\end{array}$$

The smallest possible positive function g(t) is explicitly given in terms of the semiconvexity and semiconcavity parameters of V and S 0, respectively. The optimal g does not depend on the viscosity parameter \({\varepsilon >0 }\) . The potential V and the initial function S 0 are allowed to grow quadratically.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    MATH  MathSciNet  Google Scholar 

  2. Aronson D.G., Besala P.: Uniqueness of positive solutions of parabolic equations with unbounded coefficients. Colloq. Math. 18, 125–135 (1967)

    MATH  MathSciNet  Google Scholar 

  3. Aronson D.G., Besala P.: Parabolic equations with unbounded coefficients. J. Differential Equations 3, 1–14 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ben Moussa B., Kossioris G.T.: On the system of Hamilton–Jacobi and transport equations arising in geometrical optics. Comm. Partial Differential Equations 28, 1085–1111 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. P. Cannarsa and C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations, and optimal control, Progress in Nonlinear Differential Equations and their Applications 58, Birkhäuser Boston, MA, 2004.

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Friedman A.: Partial differential equations of parabolic type. Prentice-Hall, Inc., Englewood Cliffs, N.J. (1964)

    MATH  Google Scholar 

  8. Fujita Y.: Hessian estimates for viscous Hamilton-Jacobi equations with the Ornstein-Uhlenbeck operator. Differential Integral Equations 18, 1383–1396 (2005)

    MathSciNet  Google Scholar 

  9. Il’in A.M., Kalashnikov A.S., Olenik O.A.: Second-order linear equations of parabolic type. J. Math. Sci. (New York) 108, 435–542 (2002)

    Article  MathSciNet  Google Scholar 

  10. 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 

  11. N. V. Krylov, Lectures on elliptic and parabolic equations in Hölder spaces, Graduate Studies in Mathematics 12, American Mathematical Society, Providence, RI, 1996.

  12. O. A. Ladyz̆enskaja, V. A. Solonnikov, and N. N. Ural’ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs 23, American Mathematical Society, Providence, R.I. 1967.

  13. P.-L. Lions, Generalized solutions of Hamilton–Jacobi equations, Research Notes in Mathematics 69, Pitman (Advanced Publishing Program), Boston–London, 1982.

  14. Souplet P., Zhang Q.S.: Global solutions of inhomogeneous Hamilton-Jacobi equations. J. Anal. Math. 99, 355–396 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. T. Strömberg, On a viscous Hamilton–Jacobi equation with an unbounded potential term, submitted.

  16. T. Strömberg, A system of the Hamilton–Jacobi and the continuity equations in the vanishing viscosity limit, submitted.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Luleå University of Technology, 971 87, Luleå, Sweden

    Thomas Strömberg

Authors
  1. Thomas Strömberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas Strömberg.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Strömberg, T. Semiconcavity estimates for viscous Hamilton–Jacobi equations. Arch. Math. 94, 579–589 (2010). https://doi.org/10.1007/s00013-010-0132-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-010-0132-2

Mathematics Subject Classification (2000)

Keywords

Search

Navigation