Annali di Matematica Pura ed Applicata

, Volume 145, Issue 1, pp 385–405 | Cite as

On 385-01385-01385-01regularity of the gradient of solutions of degenerate parabolic systems

  • Michael Wiegner
Article

Summary

We consider weak solutions u∈Lp([0,T], Wp1(Ω))∩([0,T],L2(Ω))of the degenerate parabolic (model) -system
$$\frac{{\partial u^i }}{{\partial t}} - div (|\nabla u|^{p - 2} \nabla u^i ) = 0 on \Omega \subset R^N , 1 \mathbin{\lower.3ex\hbox{$\buildrel<\over{\smash{\scriptstyle=}\vphantom{_x}}$}} i \mathbin{\lower.3ex\hbox{$\buildrel<\over{\smash{\scriptstyle=}\vphantom{_x}}$}} m with p > 2.$$
By local techniques it is proved, using sequences of time-space cylinders, which are adjusted to the alternative whether one is at a point of degeneracy or not, that the spatial gradient of u is α- Höldercontinuous on compact subsets of Ω× [0,T] with some α which depends only on N and p.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    N. D. Alikakos -L. C. Evans,Continuity of the gradient of solutions of certain degenerate parabolic equations, J. Math. Pures et Appl.,62 (1983), pp. 253–268.Google Scholar
  2. [2]
    N. D. Alikakos -R. Rostamian,Gradient estimates for degenerate diffusion equations I, Math. Ann.,259 (1982), pp. 53–70.Google Scholar
  3. [3]
    L. A. Caffarelli -L. C. Evans,Continuity of the temperature in the two-phase Stefan problem, Archive Rat. Mech. Anal.,81 (1983), pp. 199–220.Google Scholar
  4. [4]
    E. DiBenedetto,Continuity of weak solutions to a general porous medium equation, In- diana Univ. Math. J.,32 (1983), pp. 83–118.Google Scholar
  5. [5]
    E. DiBenedetto -A. Friedman,Regularity of solutions of nonlinear degenerate parabolic systems, Journal für Reine und Angew. Math.,349 (1984), pp. 83–128.Google Scholar
  6. [6]
    M. Giaquinta,Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, Princeton, New Jersey, 1983.Google Scholar
  7. [7]
    O. A. Ladyzhenskaya -N. N. Ural'tseva,Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.Google Scholar
  8. [8]
    O. A.Ladyzhenskaya - N. A.Solonnikov - N. N.Ural'tseva,Linear and Quasilinear Equations of Parabolic Type, Providence, R.I., 1968.Google Scholar
  9. [9]
    P. Tolksdorf,Everywhere regularity for some quasilinear systems with a lack of ellipticity, Ann. Mat. Pura Appl.,84 (1983), pp. 241–266.Google Scholar
  10. [10]
    K. Uhlenbeck,Regularity for a class of nonlinear elliptic systems, Acta Math.,138 (1977), pp. 219–240.Google Scholar
  11. [11]
    M. Wiegner,Über die Regularität schwacher Lösungen gewisser elliptischer Systeme, Manuscripta math.,15 (1975), pp. 365–384.Google Scholar
  12. [12]
    M.Wiegner,Das Existenz- und Regularitätsproblem bei Systemen nichtlinearer elliptischer Differentialgleichungen, Habilitationsschrift, Bochum, 1977.Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1985

Authors and Affiliations

  • Michael Wiegner
    • 1
  1. 1.Institut für Mathematik der Universität BayreuthBayreuthBRD

Personalised recommendations