Limit of a sequence of functions in the Sobolev space

Question

Let $\mathbb{D} \subset \mathbb{R}^2$ be the unit disk and consider the Dirichlet problem of the Laplace equation $\Delta u = 0$ in $\mathbb{D}$ with $u = 0$ on $\partial \mathbb{D}$. Then we know that the solution to this problem is $u \equiv 0$ and that it minimizes the Dirichlet energy defined as \begin{gather} E[u] := \int_{\mathbb{D}} | \nabla u|^2 \end{gather}

Construct a sequence of functions $\{u_k\} \subset C(\overline{\mathbb{D}})$ satisfying the following conditions

1. $u_k$ is piecewise smooth and $u_k|_{\mathbb{\partial D}}=0$ $\forall\ k \in \mathbb{N}$
2. $E[u_k] \to 0$ as $k \to \infty$
3. $u_k$ diverges as $k \to \infty$ in a set that is dense in $\mathbb{D}$

Then show that $u_k \to 0$ in the Sobolev space $H^1(\mathbb{D})$

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Ok, I have constructed a sequence satisfying the aforementioned properties but I cannot understand how can $u_k \to 0$ in $H^1(\mathbb{D})$. I understand that the Sobolev space is equiped with the norm \begin{align} \|u\|_{H^1} &:= \Big( \int_{\mathbb{D}} (|\nabla u |^2 + |u|^2)\Big)^{1/2}\\ \|u\|^2_{H^1} &= \|\nabla u\|^2_{L^2(\mathbb{D})} + \|u\|^2_{L^2(\mathbb{D})} \end{align} and that by Property 2 we have $\|\nabla u_k\|_{L^2(\mathbb{D})} \to 0$ as $k \to \infty$ but still $u_k$ diverges in a dense subset of $\mathbb{D}$ (Property 3) and thus we should have $\|u_k\|_{L^2(\mathbb{D})} \to \infty$ as $k \to \infty$ which means that $u \notin H^1(\mathbb{D})$.

Could you please explain me that? Also, where am I wrong in the above considerations?

Thanks!

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Update After @daw's answer I want to clarify that I constructed a sequence diverging in an uncountable dense subset. Does anything on the above declaration of the problem forbids me to do that? Or is it softly implied that the subset must be countable?

Answer 1

A dense subset still can have measure zero (think of points with rational coordinates), so your conclusion: property (3) implies $\|u_k\|_{L^2}\to\infty$ is not valid. Also it would contradict the Poincare inequality.

The construction of $u_k$ would be something like $$ u_k = \frac1{k^2} \sum_{i=1}^k \phi_i, $$ where $\phi_k \in H^1(\Omega)\setminus L^\infty(\Omega)$, $\|\phi_k\|_{H^1}=1$ with singularity at some $q_k$, where $(q_k)$ is dense in $\mathbb D$.

In order to get an approximation by continuous functions, we can approximate $\phi_k$ by $\phi_{k,n}$ with $\phi_{k,n}\to \phi_k$ and $\|\phi_{k,n}\|_{H^1}\le 2$. Then set $$ u_n = \frac1{n^2} \sum_{i=1}^n \phi_{i,n}, $$