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Norm equivalence between Sobolev space and L_2

  1. Dec 13, 2011 #1
    Hello! I've found this paper, wherein page 33 states that the reverse Poincaré inequality gives

    [tex]\forall v \in H^1_0(\Omega) , \|v\|_{L^2(\Omega)} \leq C(\Omega) \|\nabla v\|_{L^2(\Omega)} [/tex]

    This I can follow - it gives a norm equivalence between the norm of a vector and the gradient of its gradient (clumsily expressed, I know). However, just a little bit later the paper states that

    [tex]\|u\|_{H_0^1(\Omega)} = \|\nabla u\|_{L^2(\Omega)}[/tex]

    That is, those two norms are equivalent. This is what I don't understand - I don't understand the jump from the L^2-norms in the first statement to the Sobolev norm in the second. Any help in understanding this would be helpful.
  2. jcsd
  3. Dec 16, 2011 #2
    I think L2 is also in a Hilbert space. And if you take the gradient of a function, the result resides in a lower Hilbert Space.
    So I think the problem reduces to finding out which H corresponds to L2.
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