Convergence of a sequence of Hölder continuous functions with respect to the Sobolev norm

Question

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain, $W_0^{1,2}(\Omega)$ be the Sobolev space and $C^{2,\alpha}$ be the Hölder space for some $\alpha\in (0,1]$. Suppose $(u_k)_{k\in\mathbb N}\subseteq C^{2,\alpha}(\Omega)$ converges to $u\in W_0^{1,2}(\Omega)$ with respect to the Sobolev norm and converges to $\tilde{u}\in C^0(\overline\Omega)$ with respect to the supremum norm. Why do we need to have $$u=\tilde{u}\;,$$ i.e. the representatives in $u$ are almost everywhere equal to $\tilde{u}$?

Answer 1

If you note $\|\cdot \|_\infty$ the sup ess norm, you have :

$$\| u - \tilde{u} \|_{\infty} \leq \| u - u_k\|_{\infty}+ \| u_k - \tilde{u} \|_{\infty}$$

Now let $\epsilon > 0$,

• As $u_k \to \tilde{u}$ uniformly, there exist $k_1$ such that $\forall k > k_1, \ \| \tilde{u} - u_k\|_{\infty} < \frac{\epsilon}{2}$

• As $u_k \to u$ in $W^{1,2}_0$, it converge in $L^2$, so there exist a subsequence $u_{i_n}$ that converge a.e to $u$. It means that there exists $i_{n_0} > k_1$ such that $\| u - u_{i_{n_0}} \|_{\infty} < \frac{\epsilon}{2}$

So, no matter what $\epsilon>0$ you take,

$$\| u - \tilde{u} \|_{\infty} \leq \| u - u_{i_{n_0}} \|_{\infty} + \| \tilde{u} - u_{i_{n_0}} \|_{\infty} \leq \frac{\epsilon}{2}+ \frac{\epsilon}{2} \leq \epsilon $$

This imply that $\| u - \tilde{u} \|_{\infty} = 0$