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Hölder Estimates for the Gradient

  • David Gilbarg
  • Neil S. Trudinger
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 224)

Abstract

In this chapter we derive interior and global Hölder estimates for the derivatives of solutions of quasilinear elliptic equations of the form
$$ Qu = a^{ij} \left( {x,u,Du} \right)D_{ij} u + b\left( {x,u,Du} \right) = 0 $$
(12.1)
in a bounded domain Ω. From the global results we shall see that Step IV of the existence procedure described in Chapter 10 can be carried out if, in addition to the hypotheses of Theorem 10.4, we assume that either the coefficients a ij are in \(C^1(\bar{\Omega}\times {\rm R} \times {\rm R}^n)\) or that Q is of divergence form or that n = 2. The estimates of this chapter will be established through a reduction to the results of Chapter 8, in particular to Theorems 8.18, 8.24, 8.26 and 8.29.

Keywords

Dirichlet Problem Divergence Form Quasilinear Elliptic Equation Boundary Estimate Interior Estimate 
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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1977

Authors and Affiliations

  • David Gilbarg
    • 1
  • Neil S. Trudinger
    • 2
  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.Department of Pure MathematicsAustralian National UniversityCanberraAustralia

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