Global and Interior Gradient Bounds

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


In this chapter we are mainly concerned with the derivation of apriori estimates for the gradients of C 2(Ω) 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 terms of the gradients on the boundary ∂Ω and the magnitudes of the solutions. The resulting estimates facilitate the establishment of Step III of the existence procedure described in Section 10.3. On combination with the estimates of Chapters 9, 12 and 13, they yield existence theorems for large classes of quasilinear elliptic equations including both uniformly elliptic equations and equations of similar form to the prescribed mean curvature equations (9.7). Since the methods of this chapter involve the differentiation of equation (14.1), our hypotheses will generally require structural conditions to be satisfied by the derivatives of the coefficients a ij , b. In Section 14.4 we shall see that these derivative conditions can be relaxed somewhat for equations in divergence form, where different types of arguments are appropriate.


Structure Condition Elliptic Equation Dirichlet Problem Quasilinear Elliptic Equation Interior Estimate 
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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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