Advertisement

Annali di Matematica Pura ed Applicata

, Volume 130, Issue 1, pp 131–176 | Cite as

Continuity of weak solutions to certain singular parabolic equations

  • Emmanuele Di Benedetto
Article

Keywords

Weak Solution Parabolic Equation Singular Parabolic Equation 
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.

Sunto

Le equazioni paraboliche singolari (1.1)nella introduzione, si presentano come modelli di una classe generale di fenomeni di diffusione con cambio di fase. Le soluzioni deboli sono trovate in senso globale come classi di equivalenza in certi spazi di Sobolev. In questo lavoro si dimostra che le soluzioni deboli ammettono delle rappresentanti continue nell'interno del dominio di definizione. Si danno anche delle condizioni di continuità fino alla frontiera.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    H.Brèzis,On some degenerate non-linear parabolic equation, Proceedings AMS, Vol. I, XVII (1968).Google Scholar
  2. [2]
    L.Caffarelli - L. C.Evans,Continuity of the temperature in the two-phase Stefan problem, (to appear).Google Scholar
  3. [3]
    L.Caffarelli - A.Friedman,Continuity of the density of a gas flow in a porous medium, Trans. Amer. Math. Soc. (to appear).Google Scholar
  4. [4]
    L.Caffarelli - A.Friedman,Regularity of the free boundary of a gas flow in an n-dimensional porous medium, Indiana Univ. Math. J.,29, no. 3 (1980).Google Scholar
  5. [5]
    J. R.Cannon - E.Di Benedetto,On the existence of solution of boundary value problems in Fast Chemical reactions, Bollettino U.M.I., (5),15-B (1978).Google Scholar
  6. [6]
    J. R.Cannon - E.Di Benedetto,On the existence of weak solutions to an n-dimensional Stefan problem with non-linear boundary conditions, SIAM J. on Math. Analysis,11, no. 4 (1980).Google Scholar
  7. [7]
    J. R.Cannon - E.Di Benedetto - G. H.Knightly,The Stefan problem with convection: the non-steady state case, (in preparation).Google Scholar
  8. [8]
    J. R.Cannon - C. D.Hill,On the movement of a chemical reaction interface, Indiana Math. J.,20 (1970).Google Scholar
  9. [9]
    J. R.Cannon - A.Fasano,Boundary value multidimensional problems in fast chemical reactions, Archive for Rat. Mech. and Analysis,53, no. 1 (1973).Google Scholar
  10. [10]
    J. R.Cannon - D. B.Henry - D. B.Kotlow,Classical solutions of the one-dimensional two-phase Stefan problem, Annali di Mat. Pura e Appl., (IV),107 (1976).Google Scholar
  11. [11]
    E.De Giorgi,Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino, Cl. Sc. Fis. Mat. Nat., (3),3 (1957).Google Scholar
  12. [12]
    E.Di Benedetto,Regularity properties of the solution of an n-dimensional two-phase Stefan problem, Bollettino U.M.I. (to appear).Google Scholar
  13. [13]
    E.Di Benedetto - R. E.Showalter,Implicit degenerate evolution equations and applications, SIAM J. on Math. Anal., Vol.12, no. 5 (1981).Google Scholar
  14. [14]
    A. Fasano -M. Primicerio,Partially saturated porous media, J. Inst. Maths. Applics.,23 (1979), pp. 503–517.Google Scholar
  15. [15]
    A.Fasano - M.Primicerio - S.Kamin,Regularity of weak solutions of one-dimensional two-phase Stefan problem, Annali di Mat. Pura e Appl., (IV),115 (1977).Google Scholar
  16. [16]
    A.Friedman,The Stefan problem in several space variables, Trans. Amer. Mat. Soc.,132 (1968).Google Scholar
  17. [17]
    A. Friedman,Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J. (1964).Google Scholar
  18. [18]
    O. A.Ladyzenskaja - V. A.Solonnikov - N. N.Ural'ceva,Linear and Quasi-linear Equations of Parabolic Type, Amer. Math. Soc. Transl. Math. Mono,23, Providence, R.I. (1968).Google Scholar
  19. [19]
    O. A. Ladyzenskaja -N. N. Ural'ceva,Linear and Quasi-linear Elliptic Equations, Academic Press, New York (1968).Google Scholar
  20. [20]
    J. L. Lions,Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod Gauthier-Villars, Paris (1969).Google Scholar
  21. [21]
    J. L. Lions -E. Magenes,Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, Berlin, London and New York (1972).Google Scholar
  22. [22]
    S. N.Kruzkov,A priori estimates for generalized solutions of second-order elliptic and parabolic equations, Soviet Math. (1963).Google Scholar
  23. [23]
    S. N.Kružkov,A priori estimates of solutions of linear parabolic equations and of boundary value problems for a certain class of quasi-linear parabolic equations, Doklady Akad. Nauk,150 (1963).Google Scholar
  24. [24]
    S. N.Kružkov,Results concerning the nature of the continuity of solutions of parabolic equations and some of their applications, Matematicheskie Zametki,6, no. 1 (1969).Google Scholar
  25. [25]
    S. N.Kružkov - S. M.Sukorjanskii,Boundary value problems of systems of equations of two-phase porous flow type: statement of the problems, questions of solvability, justification of approximate methods, Math. USSR Sbornik,33, no. 1 (1977).Google Scholar
  26. [26]
    J. Moser,A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math.,13 (1960), pp. 457–468.Google Scholar
  27. [27]
    O. A.Oleinik - A. S.Kalashnikov -Chzhou Yui-Lin,The Cauchy problem and boundary problems for equations of the type of non-stationary filtration, Izvestija Akademii Nauk SSSR Ser. Mat.,22 (1958).Google Scholar
  28. [28]
    L.Rubinstein,The Stefan Problem, A.M.S. Translations of Mathematical Monographs, Vol.27, Providence, R.I. (1971).Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1982

Authors and Affiliations

  • Emmanuele Di Benedetto
    • 1
  1. 1.Madison

Personalised recommendations