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manuscripta mathematica

, Volume 8, Issue 1, pp 59–67 | Cite as

Über quasilineare elliptische Differentialgleichungen in der Ebene

  • Wolf von Wahl
Article

Abstract

This paper deals with a-priori estimates for Dirichlet's problem for quasilinear elliptic equations
$$a_{ij} (x,u, \triangledown u)u_{x_i x_j } = f(x,u, \triangledown u)$$
in the plane. We give an a-priori estimate for the C2+α-norms of all solutions under the following assumptions only: The principal part is uniformly elliptic, f has quadratical growth with respect to ∇u, aij and f are Hölder-continuous and an a-priori estimate for sup|u| is known.

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Literatur

  1. [1]
    AGMON, S., DOUGLAS, A., and NIRENBERG, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Commun. Pure Appl. Math. 12, 623–727 (1959).Google Scholar
  2. [2]
    LADYŽENSKAJA, O.A., et URAL'CEVA, N.N.: Equations aux derivées partielles de type elliptique. Paris: Dunod 1968Google Scholar
  3. [3]
    TOMI, F.: Über semilineare elliptische Differentialgleichungen zweiter Ordnung, Math. Z. 111, 350–366 (1969)Google Scholar
  4. [4]
    VON WAHL, W.: A-priori Schranken für semilineare und quasilineare parabolische Differentialgleichungen. Math. Z., erscheint demnächst.Google Scholar

Copyright information

© Springer-Verlag 1973

Authors and Affiliations

  • Wolf von Wahl
    • 1
  1. 1.Mathematisches InstitutUniversität Bonn53 Bonn

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