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Let $H = H_{0}^{1}(\Omega)$ where $\Omega$ is a bounded domain in $R^N$ whose boundary $\partial\Omega$ is a smooth manifold. We know that the embedding $$H\hookrightarrow L^s(\Omega)$$ is compact for each $s\in [1, 2^*)$ and continuous for each $s\in [1, 2^*]$

Suppose that $\{u_n\}$ is a sequence in $H$ such that $u_n\rightharpoonup u$. Then $\{u_n\}$ is bounded, so by the compact embedding there exists a subsequence $\{u_{n_k}\}$ such that $u_{n_k}\rightarrow u_0$ in $L^s(\Omega)$ for $s\in [1, 2^*)$

How do we know that $u_0 = u$?

  • 0
    Can you define what you mean exactly by $u_n \hookrightarrow u$? Is that weak convergence?2011-12-12
  • 0
    For any continuous and linear functional $f$ on $H$, $u_n\rightharpoonup u$ in $H$ means that $f(u_n)\rightarrow f(u)$.2011-12-12
  • 0
    $H\hookrightarrow L^s(\Omega)$, i.e. there is a identity operate $I$ and a positive constant $M$ such that $$\|u\|_L\leq M\|u\|_H$$ for any $u\in H$.2011-12-12

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