Coincidence of a weak limit and a uniform limit in Sobolev spaces

Question

Let $\Omega$ be a nice open, bounded domain in $\mathbb{R}^n$. (e.g suppose $\bar \Omega$ is a smooth manifold with boundary).

Let $f_n \in W^{1,p}(\Omega),p>n$ and suppose that:

1. $f_n \rightharpoonup f$ in $W^{1,p}(\Omega)$.

2. $f_n $ converges uniformly to a function $g$. (Since $p>n$ the $f_n$ are continuous, so this is well-defined).

How can I conclude $f=g$ almost everywhere?

Answer 1

Since $\Omega$ is bounded and $f_n$ converges uniformly to $g$, we have $$ \lim_{n\to\infty}\int_\Omega f_n\,\phi=\int_\Omega g\,\phi $$ for all test functions $\phi$. Thus, $f_n$ converges weekly to $g$. Since weak limits are unique, $f=g$.

Answer 2

This is an elaboration on Julian Aguirre's answer.

Since $f_n$ weakly converges to $f$ in $W^{1,p}$ it follows that $f_n$ weakly converges to $f$ in $L^p$.

(Since $W^{1,p} \subseteq L^p \Rightarrow (L^p)^* \subseteq (W^{1,p})^*$).

However, since $f_n$ converges uniformly $g$ and , it follows that $f_n \to g$ weakly in $L^p$: Indeed, let $h \in L^q$. Then

$$ \lim_{n\to\infty}\int_\Omega f_n\,h=\int_\Omega g\,h.$$

Indeed $$ |\int_\Omega f_n\,h-\int_\Omega g\,h| \le \int_\Omega |f_n-g|\,|h|\le ||f_n-g||_{sup}\|h\|_{L^1} \to 0, $$

where we have used the fact $\Omega$ is bounded, so $h \in L^q \rightarrow h \in L^1 $.

Now, Since weak limits are unique, $f=g$.