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Sobolev Spaces

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

Abstract

To motivate the theory of this chapter we now consider a different approach to Poisson’s equation from that of Chapter 4. By the divergence theorem (equation (2.3)) a C2(Q) solution of Δu=f satisfies the integral identity
$$ \int\limits_{\Omega } {Du \cdot D\varphi dx = - \int\limits_{\Omega } {f\varphi dx} } $$
(7.1)
for all φC 0 1 ∈ (Ω). The bilinear form
$$ \left( {u,\varphi } \right) = \int\limits_{\Omega } {Du \cdot D\varphi dx} $$
(7.2)
is an inner product on the space C 0 1 (Ω) and the completion of C 0 1 (Ω) under the metric induced by (7.2) is consequently a Hubert space, which we call W 0 1,2 (Ω).

Keywords

Banach Space Sobolev Space Chain Rule Sobolev Inequality Interpolation Inequality 
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 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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