Skip to main content
SpringerLink
Log in

The continuity or stability method for nonlinear elliptic and parabolic equations and systems

  • Published:
Rendiconti del Seminario Matematico e Fisico di Milano Aims and scope Submit manuscript

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

References

  • [Ca] Campanato S.,On the nonlinear parabolic systems in divergence form. Hölder continuity and partial Hölder continuity of the solutions. Ann. Mat. Pura Appl. 137, 83–122, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  • [FK] Fujita H. and Kato T.,On the Navier-Stokes initial value problem, I. Arch. Rat. Mech. Anal. 16, 269–315 1964.

    Article  MathSciNet  MATH  Google Scholar 

  • [G] Gerhardt C.,Stationary solutions to the Navier-Stokes equations in dimension four. Math. Z. 165, 193–197, 1970.

    Article  MathSciNet  Google Scholar 

  • [GM1] Giaquinta M. and Modica L.,Non linear systems of the type of the stationary Navier-Stokes system. J. Reine Angew. Math. 330, 173–214, 1982.

    MathSciNet  MATH  Google Scholar 

  • [GM2] Giaquinta M. and Modica L.,Regularity results for some classes of higher order non linear elliptic systems. J. Reine Angew. Math. 311/312, 145–169, 1979.

    MathSciNet  Google Scholar 

  • [GS] Giaquinta M. and Struwe M.,An optimal regularity result for a class of quasilinear parabolic systems. Manuscripta math. 36, 223–239, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  • [GT] Gilbarg D. and Trudinger N.S.,Elliptic Partial Differential Equations of Second Order, 2nd Edition. Springer: Berlin, Heidelberg, New York, Tokyo,—1983.

    MATH  Google Scholar 

  • [GrvW] Grunau H.C. and Wahl W. von,Regularity of Weak Solutions of Semilinear Parabolic Systems of Arbitrary Order. To appear. Preprint, 1991.

  • [H] Hamilton R.S.,Harmonic Maps of Manifolds with Boundary. Lecture Notes in Mathematics 471.— Springer: Berlin, Heidelberg, New York, 1975.

    MATH  Google Scholar 

  • [HKW] Hildebrandt S., Kaul H. and Widman H.O.,An Existence Theorem for Harmonic Mappings of Riemannian Manifolds. Acta Math. 138, 1–16, 1977.

    Article  MathSciNet  MATH  Google Scholar 

  • [J] Jäger W.,Ein Maximumprinzip fur ein System nichtlinearer Differentialgleichungen. Nachr. Ak. d. Wiss. Gottingen, II. Mathematisch-Physikalische Klasse, 1–29, 1978.

  • [JK] Jäger W. and Kaul H.,Uniqueness and Stability of Harmonic Maps and their Jacobi Fields. Manuscripta math. 28, 269–291, 1979.

    Article  MathSciNet  MATH  Google Scholar 

  • [Jo] Jost J.,Ein Existenzbeweis für harmonische Abbildungen, die ein Dirichletproblem lösen, mittels der Methode des Wärmeflusses. Manuscripta math. 34, 17–25, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  • [K] Kilimann N.,Ein Maximumprinzip fur nichtlineare parabolische Systeme, Math. Z. 171, 227–230, 1981.

    Article  MathSciNet  Google Scholar 

  • [LU] Ladyžhenskaja O.A. and Ural'tseva N.N.,Linear-and Quasilinear Elliptic Equations, Academic Press: New York and London, 1968.

    Google Scholar 

  • [LSU] Ladyžhenskaja O.A., Solonnikov V.A. and Ural'tseva N.N.,Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs 23. American Mathematical Society: Providence, R.I., 1968.

    Google Scholar 

  • [LSvW] Lunardi A., Sinestrari E. and Wahl W. von,A semigroup approach to the time dependent parabolic initial boundary value problem. To appear in: Differential and Integral Equations. Preprint, 1989.

  • [L] Luckhaus S.,Existence and regularity of weak solutions to the Dirichlet problem for semilinear elliptic systems of higher order. J. Reine Angew. Math. 306, 192–207, 1979.

    MathSciNet  MATH  Google Scholar 

  • [S] Serrin J.,The initial value problem for the Navier-Stokes equations. University of Wisconsin Press: Madison, Wisconsin, Non-linear Problems, Proceedings of a Symposium, Madison 1962 (R. Langer ed.), 69–98, 1963.

  • [SvW] Sohr H., Wahl W. von,On the Singular Set and the Uniqueness of Weak Solutions of the Navier-Stokes Equations. Manuscripta math. 49, 27–59, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  • [T] Tomi F.,Über semilineare elliptische Differentialgleichungen zweiter Ordnung. Math. Z. 111, 350–366, 1969.

    Article  MathSciNet  MATH  Google Scholar 

  • [W1] Wahl W. von,A-priori Schranken für semilineare und quasilineare parabolische Differentialgleichungen. Math. Z. 128, 93–108, 1972.

    Article  MathSciNet  MATH  Google Scholar 

  • [W2] Wahl W. von,Über nichtlineare elliptische Gleichungen in der Ebene. Manuscripta math. 8, 59–67, 1973.

    Article  MathSciNet  MATH  Google Scholar 

  • [W3] Wahl W. von,Existenzsätze für nichtlineare elliptische Systeme. Nachr.Ak.d. Wiss. Gottingen. II. Mathematisch-Physikalische Klasse, 131–177, 1981.

  • [W4] Wahl W. von,Klassische Lösbarkeit im Grossen fur nichtlineare parabolische Systeme und das Varhalten der Lösungen für t→∞. Nachr.Ak.d. Wiss. Gottingen, II. Mathematisch-Physikalische Klasse, 131–177, 1981.

  • [W5] Wahl W. von,Global Solutions to Evolution Equations of Parabolic Type. Lecture Notes in Mathematics 1223, Proceedings of the Conference on Differential Equations in Banach Spaces, Bologna 1985 (A. Favini and E. Obrecht editors), 254–266, 1986.

  • [W6] Wahl W. von,The Equations of Navier-Stokes and Abstract Parabolic Equations. Vieweg. Braunschweig, Wiesbaden, 1985.

    Google Scholar 

  • [W7] Wahl W. von,Extension of a result of Ladyžhenskaja and Ural'tseva concerning second order parabolic equations to equations of arbitrary order. Ann. Pol. Math. 41, 63–72, 1983.

    MATH  Google Scholar 

  • [Wi1] Wiegner M.,Ein optimaler Regularitätssatz fur schwache Lösungen gewisser elliptischer Systeme. —Math. Z. 147, 21–28, 1976.

    Article  MathSciNet  MATH  Google Scholar 

  • [Wi2] Wiegner M.,On the Asymptotic Behaviour of Solutions of Non-linear Parabolic Equations, Math. Z. 188, 3–22, 1984.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universität Bayreuth, Bayreuth, Germany

    Wolf von Wahl

Authors
  1. Wolf von Wahl
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Dedicated to prof. L. Amerio

Conferenza tenuta il 30 settembre 1992

Rights and permissions

Reprints and permissions

About this article

Cite this article

von Wahl, W. The continuity or stability method for nonlinear elliptic and parabolic equations and systems. Seminario Mat. e. Fis. di Milano 62, 157–183 (1992). https://doi.org/10.1007/BF02925442

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02925442

Keywords

Search

Navigation