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Sobolev Spaces

  • David Gilbarg
  • Neil S. Trudinger
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, 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 C 2(Ω) solution of ⊿u = f satisfies the integral identity
$$ \int\limits_\Omega {Du \cdot D\phi dx = - \int\limits_\Omega {f\phi dx} } $$
(7.1)
for all φ ∊ C 0 1(Ω). The bilinear form
$$ \left( {u,\phi } \right) = \int\limits_\Omega {Du} \cdot D\phi dx $$
(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 Hilbert space, which we call W 0 1,2(Ω).

Keywords

Banach Space Sobolev Space Chain Rule Sobolev Inequality Difference Quotient 
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 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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