Weak lower semincontinuity of a functional with weak lower semicontinuity of $W^{1,2}$-norm

Question

Let $\Omega\subset \mathbb{R}^n$ be open and bounded. Let $(f_n)_{n\in \mathbb{N}}\subset W^{1,2}(\Omega)$ be bounded. Then there is a subsequence $(f_{n_k})_{k\in \mathbb{N}}$ which converges weakly in $W^{1,2}(\Omega)$. Consider $g:W^{1,2}(\Omega)\to \mathbb{R}$, \begin{align*} g(f)=\int\limits_{\Omega} a|f|^2+ b|\nabla f|^2 dx \end{align*} where $a\neq b$, $a,b\in \mathbb{R}$. Can I show the weak lower semincontinuity of $g$ via the weak lower semincontinuity of the $||\cdot||_{W^{1,2}(\Omega)}$-norm? I don't see how to show the lower semincontinuity with $a\neq b$. I would be thankful for a helpful comment.

Answer 1

I would not use the lsc of the norm. Instead, since we have $(f_{n_k})_{k\in \mathbb{N}}$ which converges in $W^{1,2}(\Omega)$, then we have even (strong) continuity of the functional. Split the integral and look at the terms separately. For example the second term:

$$b\int\limits_{\Omega} |\nabla f_{n_k}|^2 dx \rightarrow b\int\limits_{\Omega} |\nabla f|^2 dx$$

by definition of convergence in $W^{1,2}(\Omega)$ (which implies convergence in $L^2(\Omega)$ of both $f$ and the weak derivatives of $f$).

Finally:

\begin{align*} g(f)=\int\limits_{\Omega} a|f|^2+ b|\nabla f|^2 dx ​= \lim_k g(f_{n_k})\end{align*}

Then, if you prove convexity of the functional, by an application of Mazur’s theorem you get the weak continuity of the functional $g$.

Answer 2

In order to show convexity, we have to assume $a,b\geq 0$. The convexity can then be shown by using Young's inequality. Further the strong lsc is easy since if $f_n\to f$ in $W^{1,2}(\Omega)$, then $f_n\to f$ in $L^2$ and $f_n\to f$ in $H_0^1$. Hence \begin{align*} a\int\limits_{\Omega} |f_n|^2 dx \to a\int\limits_{\Omega} |f|^2dx, \end{align*} and \begin{align*} b\int\limits_{\Omega} |\nabla f_n|^2 dx \to b\int\limits_{\Omega} |\nabla f|^2dx. \end{align*} We have $\lim\limits_{n\to \infty} g(f_n)=g(f)$ and hence $g$ is strongly lsc. Further since $g$ is strongly lsc and convex, it is also weak lsc. This was what you suggested, right ?