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The question is about the dual function space of Sobolev space $W^{1,p}​$.

Let $\Omega$ be a bounded Lipchitz domain of $\mathbb{R}^3$. For purpose of simplicity of argument, domain $\Omega$ can be even selected as a cube.

Let p,q be the pair such that $1/p+1/q=1,q>3, 0<p<3/2$.

Given $v \in W^{1,q}(\Omega) $, $v$ can be regarded a functional on $W^{1,p}$, i.e., $v\in {W^{-1,p}}$. Then, how to prove that there exists $C>0$ such that $$ \|v\|_{{W^{-1,p}} } = \sup_{0 \neq ||u||_{1,p} \leq 1} \int_\Omega \nabla u \cdot \nabla v + uv ~\mbox{d}\Omega \geq C ||v||_{1,q} \quad (0<C ) $$ Does the constant $C$ equal to $1$?

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I am not totally sure for this problem. But I felt that you should first prove that $W^{-1,p}(\Omega)=(W^{1,p}(\Omega))^*$. But this can follow immediately by definition, definition itself. If this is the case, then simply use the Hahn-Banach theorem.

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  • $\begingroup$ I think you want a $p’$ there (the Holder conjugate) $\endgroup$ – David Bowman May 29 '18 at 3:54
  • $\begingroup$ I agree with you that $W^{-1,p}(\Omega)=(W^{1,q}(\Omega))^*$, which is from the definition. Then, next, how to continue the proof with Han-Banach theorem. I am sorry that maybe it is too trivial, but I cannot figure out the proof. $\endgroup$ – Xuefeng LIU May 29 '18 at 14:16

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