Showing the energy of Dirichlet problem is lower semicontinuous.

Question

I am proving the existence of minimizers in $H_0^1(\Omega)$ of $$E[u] = \int |\nabla u|^2 -\int fu.$$

After showing that it is bounded, we pick a minimizing sequence $u_k \rightarrow u$ weakly in $H_0^1(\Omega)$ and stronlgy in $L^2(\Omega)$. Now I need to claim that $$E[u]\leq \liminf E[u_k].$$

This is the fact that $E$ is lower semicontinuous, but I am stuck at showing this directly. By weak convergence, we have: $$\int |\nabla u||\nabla u_k| +\int u u_k \rightarrow \int |\nabla u|^2 +\int u^2.$$ And by $L^2$ convergence, the second term converges, so $$\int |\nabla u||\nabla u_k|\rightarrow \int |\nabla u|^2 .$$

How do we go from here to prove the lower semicontinuity?

Answer 1

Since $u_k \rightarrow u$ weakly, we have: \begin{equation} \int|\nabla u_k||\nabla u| + \int u_k u \rightarrow \int |\nabla u|^2 + \int u^2. \end{equation} The second term converges by strong $L^2$ convergence, so we have: \begin{equation} \|\nabla u\|^2_{L^2} =\int|\nabla u|^2 \leq \liminf \int |\nabla u_k||\nabla u| \leq \liminf \|\nabla u_k\|_{L^2}\|\nabla u\|_{L^2} \end{equation} Dividing both sides by $\|\nabla u\|_{L^2}$ gives $\|\nabla u\|_{L^2} \leq \liminf\|\nabla u_k\|_{L^2}$. Therefore, \begin{align} E[u] = \frac{1}{2}\int |\nabla u|^2 - \int uf \leq \liminf \frac{1}{2} \int |\nabla u_k|^2 - \int u_kf = \liminf E[u_k] \end{align}

Answer 2

The claim $$ \int_\Omega |\nabla u_k| \cdot |\nabla u| \to \int |\nabla u|^2 $$ is wrong. Another line to argue lower semicontinuity is to recognize that $u\mapsto \int |\nabla u|^2$ is a norm on $H^1_0$, and norms are weakly lower semicontinuous.