An integral of a sequence of functions

Question

$\Omega\subset\mathbb{R}^n$ is a bounded domain with smooth boundary $\partial\Omega$. Does $$ \liminf_{k\rightarrow\infty} \int_{\Omega} \rho(u_k)\,dx \geq \int_{\Omega} \liminf_{k\rightarrow\infty} \rho(u_k)\,dx $$ hold when $\rho$ is a $C^1$ function, and the sequence of functions $\{ u_k(x) \}_{k=1}^{k=\infty}\in H^1_{0}(\Omega)$ satisfies \begin{equation} u_k \rightarrow u^{\ast} \quad \text{a.e.} \quad \text{in}\quad \Omega \quad \text{as} \quad k\rightarrow \infty? \end{equation}

Answer 1

Making Your Ad Here's example explicit: fix $x_0\in \Omega$ and let $\rho(t)= - t$. Define $u_k(x) = k^{n-1} \max(0, 1-k|x-x_0|)$. The graph of $k$ is the cone with vertex $(x_0,k^{n-1})$ and base $B(x_0,1/k)$. For large $k$ we have $u_k\in H^1_0(\Omega)$.

On one hand, $\int_{\Omega} \rho( u_k)$ is a fixed negative value, independent of $k$. On another hand, $u_k\to 0$ a.e.