Sobolev Spaces

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


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
$$ \int\limits_{\Omega } {Du \cdot D\varphi dx = - \int\limits_{\Omega } {f\varphi dx} } $$
for all φC 0 1 ∈ (Ω). The bilinear form
$$ \left( {u,\varphi } \right) = \int\limits_{\Omega } {Du \cdot D\varphi dx} $$
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).


Banach Space Sobolev Space Chain Rule Sobolev Inequality Interpolation Inequality 
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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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