If $\Omega$ is a bounded domain of $\mathbb{R}^n$ with smooth boundary and $f\in L^2(\Omega)$, and the system ($\epsilon>0$) $$-\epsilon\Delta u^\epsilon +u_t^{\epsilon}=f$$ in $\Omega$ and $$u^\epsilon=0 \text{in} \ \partial \Omega$$

has unique solution $u^\epsilon$, where $u=u(x,t): \Omega \times (0,\infty)\rightarrow \mathbb{R}$

Prove that $u^\epsilon$ converges strongly to $f$ in $L^2(\Omega)$ (Hint: We can start by proving a weak convergence)

Is it suffice to prove that $u_\epsilon$ converges strongly to $f$ in $H^{-1}$ as $\epsilon$ approaches 0.

  • $\begingroup$ what is your $u_t^\epsilon$? $\endgroup$ – xpaul Nov 29 '15 at 20:21
  • $\begingroup$ @xpaul I edited the problem, it means the partial derivative $\endgroup$ – nerd Nov 29 '15 at 20:28
  • $\begingroup$ Now it makes sense. $\endgroup$ – xpaul Nov 29 '15 at 20:29
  • $\begingroup$ Sorry for the typo @xpaul $\endgroup$ – nerd Nov 29 '15 at 20:30

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