Hölder Estimates for the Gradient

  • David Gilbarg
  • Neil S. Trudinger
Part of the Classics in Mathematics book series (volume 224)


In this chapter we derive interior and global Hölder estimates for the derivatives of solutions of quasilinear elliptic equations of the form
$$Qu = {a^{ij}}\left( {x,u,Du} \right){D_{ij}}u + b\left( {x,u,Du} \right) = 0$$
in a bounded domain Ω. From the global results we shall see that Step IV of the existence procedure described in Chapter 11 can be carried out if, in addition to the hypotheses of Theorem 11.4, we assume that either the coefficients a ij are in Cl(ΩΩ × ℝ × 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.


Dirichlet Problem Quasilinear Elliptic Equation Interior Estimate Linear Elliptic Equation Holder Estimate 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • David Gilbarg
    • 1
  • Neil S. Trudinger
    • 2
  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.School of Mathematical SciencesThe Australian National UniversityCanberraAustralia

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