Lower semicontinuity of weighted path length in BV

  • Paolo Baiti
  • Alberto Bressan
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 32)


We establish some basic lower semicontinuity properties for a class of weighted metrics in BV. These Riemann-type metrics, uniformly equivalent to the L 1 distance, are defined in terms of the Glimm interaction potential. They are relevant in the study of nonlinear hyperbolic systems of conservation laws, being contractive w.r.t. the corresponding flow of solutions.


Riemann Problem Weighted Distance Finsler Manifold Interaction Estimate Elementary Path 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Bressan, A locally contractive metric for systems of conservation laws, Ann. Scuola Norm. Sup. Pisa IV-22 (1995), 109–135.MathSciNetGoogle Scholar
  2. [2]
    A. Bressan, The semigroup approach to systems of conservation laws, Mathematica Contemporanea 10 (1996), 21–74.MathSciNetMATHGoogle Scholar
  3. [3]
    A. Bressan and R. M. Colombo, The semigroup generated by 2 x 2 conservation laws, Arch. Rational Mech. Anal. 133 (1995), 1–75.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    A. Bressan and R. M. Colombo, Unique solutions of 2 x 2 conservation laws with large data, Indiana Univ. Math. J. 44 (1995), 677–725.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    A. Bressan, G. Crasta and B. Piccoli, Well-posedness of the Cauchy problem for n x n systems of conservation laws, preprint S.I.S.S.A., Trieste, 1996.Google Scholar
  6. [6]
    A. Bressan and A. Marson, A variational calculus for discontinuous solutions of systems of conservation laws, Comm. Part. Diff. Equat. 20 (1995), 1491–1552.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    M. Crandall, The semigroup approach to first-order quasilinear equations in several space variables, Israel J. Math. 12 (1972), 108–132.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    K. Deimling, “Nonlinear Functional Analysis”, Springer-Verlag, Berlin, 1985.MATHCrossRefGoogle Scholar
  9. [9]
    L. C. Evans and R. F. Gariepy, “Measure Theory and Fine Properties of Functions”, CRC Press, 1992.Google Scholar
  10. [10]
    J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697–715.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    S. Kruzkov, First order quasilinear equations with several space variables, Math. USSR Sbornik 10 (1970), 217–243.CrossRefGoogle Scholar
  12. [12]
    P. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10 (1957), 537–566.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    J. Smoller, “Shock Waves and Reaction-Diffusion equations”, Springer-Verlag, New York, 1983.MATHCrossRefGoogle Scholar
  14. [14]
    B. Temple, No L 1-contractive metrics for systems of conservation laws, Trans. Amer. Math. Soc. 288 (1985), 471–480.MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Paolo Baiti
    • 1
  • Alberto Bressan
    • 1
  1. 1.S.I.S.S.A.TriesteItaly

Personalised recommendations