How to construct a minimizing sequence?

Question

Let ${u_k}$ be a harmonic function sequence that is continuous on a unit disk. How to construct the sequence such that,

1. $ {u_k}$ are piecewise smooth and $u_k=0 $ on the boundary
2. $ {u_k}$ make the Dirichlet energy to be 0 as $ k\to \infty $
3. $ {u_k} $ diverge in a set which is dense in the disc

is ${u_k}$ converge to 0 in $H_1(D)$ ?

Answer 1

Using Poincare inequality (as $u_k = 0$ on the boundary) you can show that $||u_k||_2$ goes to zero. This means that $u_k \to 0$ in $L^2$. As $||Du_k||_2 \to 0$, $u_k$ converges to $0$ in $H^1_0$.