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Global and Interior Gradient Bounds

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Part of the Classics in Mathematics book series (CLASSICS,volume 224)

Abstract

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

$${Q_u} = {a^{ij}}(x,u,Du){D_{ij}}u + b(x,u,Du) = 0$$
(15.1)

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 11.3. On combination with the estimates of Chapters 10,13 and 14, they yield existence theorems for large classes of quasilinear elliptic equations including both uniformly elliptic equations and equations of form similar to the prescribed mean curvature equation (10.7). Since the methods of this chapter involve the differentiation of equation (15.1), our hypotheses will generally require structural conditions to be satisfied by the derivatives of the coefficients a ij,b. In Section 15.4 we shall see that these derivative conditions can be relaxed somewhat for equations in divergence form, where different types of arguments are appropriate.

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© 2001 Springer-Verlag Berlin Heidelberg

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Gilbarg, D., Trudinger, N.S. (2001). Global and Interior Gradient Bounds. In: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics, vol 224. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-61798-0_15

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  • DOI: https://doi.org/10.1007/978-3-642-61798-0_15

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41160-4

  • Online ISBN: 978-3-642-61798-0

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