Skip to main content
SpringerLink
Log in

The Dirichlet problem associated to the relativistic heat equation

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

An Erratum to this article was published on 16 May 2015

Abstract

We prove existence and uniqueness of entropy solutions for the nonhomogeneous Dirichlet problem associated to the relativistic heat equation.

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. Amar M., De Cicco V., Fusco N.: Lower semicontinuity and relaxation results in BV for integral functionals with BV integrands. ESAIM Control Optim. Calc. Var. 14, 456–477 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs (2000)

  3. Andreu F., Ballester C., Caselles V., Mazón J.M.: The Dirichlet problem for the total variation flow. J. Funct. Anal. 180, 347–403 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Andreu F., Caselles V., Mazón J.M.: Existence and uniqueness of solution for a parabolic quasilinear problem for linear growth functionals with L 1 data. Math. Ann. 322, 139–206 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Andreu F., Caselles V., Mazón J.M.: A parabolic quasilinear problem for linear growth functionals. Rev. Mat. Iberoam. 18, 135–185 (2002)

    MATH  Google Scholar 

  6. Andreu F., Caselles V., Mazón J.M.: Erratum: a parabolic quasilinear problem for linear growth functionals. Rev. Mat. Iberoam. 24, 387–390 (2008)

    MATH  MathSciNet  Google Scholar 

  7. Andreu, F., Caselles, V., Mazón, J.M.: Parabolic quasilinear equations minimizing linear growth functionals. Progress in Mathematics, vol. 223. Birkhauser (2004)

  8. Andreu F., Caselles V., Mazón J.M.: A strong degenerate quasilinear equation: the elliptic case. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) III, 555–587 (2004)

    Google Scholar 

  9. Andreu F., Caselles V., Mazón J.M.: A strong degenerate quasilinear equation: the parabolic case. Arch. Ration. Mech. Anal. 176, 415–453 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Andreu F., Caselles V., Mazón J.M.: A strongly degenerate quasilinear elliptic equation. Nonlinear Anal. TMA 61, 637–669 (2005)

    Article  MATH  Google Scholar 

  11. Andreu F., Caselles V., Mazón J.M.: The Cauchy problem for a strong degenerate quasilinear equation. J. Eur. Math. Soc. 7, 361–393 (2005)

    Article  MATH  Google Scholar 

  12. Andreu F., Caselles V., Mazón J.M.: Some regularity results on the ‘relativistic’ heat equation. J. Differ. Equ. 245(12), 3639–3663 (2008)

    Article  MATH  Google Scholar 

  13. Andreu, F., Caselles, V., Mazón, J.M.: Some flux-limited quasi-linear elliptic equations without coercivity. Quaderni di Matematica (to appear) (University of Napoli)

  14. Andreu F., Caselles V., Mazón J.M., Moll S.: Finite propagation speed for limited flux diffusion equations. Arch. Ration. Mech. Anal. 182, 269–297 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Anzellotti G.: Pairings between measures and bounded functions and compensated compactness. Ann. di Matematica Pura ed Appl. IV(135), 293–318 (1983)

    Article  MathSciNet  Google Scholar 

  16. Anzellotti G.: The Euler equation for functionals with linear growth. Trans. Am. Math. Soc. 290, 483–500 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  17. Bénilan P., Boccardo L., Gallouet T., Gariepy R., Pierre M., Vazquez J.L.: An L 1-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Normale Superiore di Pisa, IV XXII, 241–273 (1995)

    Google Scholar 

  18. Bénilan, P., Crandall, M.G., Pazy, A.: Evolution Equations Governed by Accretive Operators. (Book in preparation)

  19. Brenier, Y.: Extended Monge-Kantorovich theory. In: Franca, M., Caffarelli, L.A., Salsa, S. (eds.) Optimal Transportation and Applications: Lectures Given at the C.I.M.E. Summer School. Lecture Notes in Math. 1813, pp. 91–122. Springer-Verlag (2003)

  20. Browder F.: Pseudo-monotone operators and nonlinear elliptic boundary value problems on unbounded domains. Proc. Natl. Acad. Sci. U.S.A. 74, 2659–2661 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  21. Chen G.-Q., Frid H.: Divergence-measure fields and hyperbolic conservation laws. Arch. Ration. Mech. Anal. 147, 89–118 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  22. Crandall M.G., Liggett T.M.: Generation of semigroups of nonlinear transformations on general Banach spaces. Am. J. Math. 93, 265–298 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  23. Dal Maso G.: Integral representation on BV(Ω) of Γ-limits of variational integrals. Manuscr. Math. 30, 387–416 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  24. Diestel, J., Uhl, J.J. Jr.: Vector Measures. Math. Surveys 15, Amer. Math. Soc., Providence (1977)

  25. Duderstadt J.J., Moses G.A.: Inertial Confinement Fusion. Wiley, New York (1982)

    Google Scholar 

  26. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press (1992)

  27. Mihalas D., Mihalas B.: Foundations of Radiation Hydrodynamics. Oxford University Press, New York (1984)

    MATH  Google Scholar 

  28. Rosenau P.: Tempered diffusion: a transport process with propagating front and inertial delay. Phys. Rev. A 46, 7371–7374 (1992)

    Article  Google Scholar 

  29. Schaefer H.H.: Banach Lattices and Positive Operators. Springer Verlag, Berlin (1974)

    MATH  Google Scholar 

  30. Schwartz, L.: Fonctions mesurables et *-scalairement mesurables, mesures banachiques majorées, martingales banachiques, et propiété de Radon-Nikodým. Sém. Maurey-Schawartz, 1974–75, Ecole Polytech., Centre de Math.

  31. Ziemer W.P.: Weakly Differentiable Functions, GTM 120. Springer Verlag, New York (1989)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departament d’Anàlisi Matemàtica, Universitat de València, Burjassot, Spain

    Fuensanta Andreu, Jose M. Mazón & Salvador Moll

  2. Departament de Tecnologia, Universitat Pompeu Fabra, Barcelona, Spain

    Vicent Caselles

Authors
  1. Fuensanta Andreu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vicent Caselles
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jose M. Mazón
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Salvador Moll
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jose M. Mazón.

Additional information

Dedicated to the memory of Fuensanta Andreu, our colleague and friend who sadly passed away on December 26th 2008.

An erratum to this article is available at http://dx.doi.org/10.1007/s00208-015-1227-7.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Andreu, F., Caselles, V., Mazón, J.M. et al. The Dirichlet problem associated to the relativistic heat equation. Math. Ann. 347, 135–199 (2010). https://doi.org/10.1007/s00208-009-0428-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-009-0428-3

Mathematics Subject Classification (2000)

Search

Navigation