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Hölder Estimates for the Gradient

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Part of the Classics in Mathematics book series (CLASSICS,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

$$Qu = {a^{ij}}\left( {x,u,Du} \right){D_{ij}}u + b\left( {x,u,Du} \right) = 0$$
(13.1)

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 C l(ΩΩ × ℝ × ℝn) 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.

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© 2001 Springer-Verlag Berlin Heidelberg

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Gilbarg, D., Trudinger, N.S. (2001). Hölder Estimates for the Gradient. 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_13

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  • DOI: https://doi.org/10.1007/978-3-642-61798-0_13

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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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