Positive Hausdorff measure and $L^{2}$ convergence.

Question

i would like to know the relation between the positive Hausdorff measure and $L^{2}$ convergence in the ((1.3) existente theorem)

Answer 1

Define the functions $v_k=u_k-u^0$ and take $R>0$ large enough so that $\mathcal{H}^{n-1}(S\cap B_R)>0$. We have that $\nabla v_k$ are bounded in $L^2(\Omega\cap B_R)$ and they vanish on $S\cap B_R$. Looking at this question (and noticing that $\Gamma$ doesn't need to be part of the boundary there), you realize that the functions $v_k$ are bounded in $H^1(\Omega)$, from which you get $u_k-u^0 \to u-u^0$ and $\nabla(u_k-u^0) \rightharpoonup \nabla(u-u^0)$ in $L^2(\Omega\cap B_R)$.

Basically, the part that you underlined is a consequence of a variant of the Poincaré inequality for $H^1_0(\Omega)$ where the functions vanish on a sufficiently fat set instead of the boundary.