Sobolev Spaces

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


Banach Space Sobolev Space Chain Rule Sobolev Inequality Difference Quotient 
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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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