This is an exercise in "Variation et optimisation des formes", chapter 3, Ex. 3.8. The preliminaries are: $D=(0,1)^2,\ f \in L^2(D),\ x_{ij}=(i/n, j/n),\ 0 $\Omega_n = D\setminus \bigcup_{0
We pick $r_n=e^{-dn^2}$.
The notation $u_\Omega^f$ is the unique solution of the problem $-\Delta u=f$, which means $ \int_\Omega fv=\int_\Omega \nabla u \nabla v,\ \forall v \in H_0^1(\Omega).$
1) Prove that there exists $u^* \in H_0^1(D)$ and a subsequence of $u_n$ which converges weakly in $H_0^1(D)$ to $u^*$ and strongly in $L^2(D)$ to the same function.
This is not very hard, and there are some theorems in the book which help.
2) This is the part where it becomes messy. Consider the squares centred at $x_{ij}$ with sides $1/n$. Consider further the circles inscribed in these squares and denote with $C_{ij}^n$ the annulus formed by the inscribed circle in the square (centred in $x_{ij}$ with radius $1/2n$) and the circle of center $x_{ij}$ and radius $r_n$.
Define the following functions $z_n \in H^1(D)$ with the properties
- $\Delta z_n=0$ on $C_{ij}^n$;
- $z_n=0$ on $B(x_{ij},r_n)$;
- $z_n=1$ elsewhere.
Determine the expression of $z_n$ for each small square (cell)[I managed to do this assuming that the solutions are radially constant around $x_{ij}$]. Here comes the part I can't get:
Prove that for each small square $-\Delta z_n=\mu_n-\nu_n$ where $\mu_n,\nu_n$ are the positive measures with support respectively the circles $C(x_{ij},1/2n)$ and $C(x_{ij},r_n)$.
The problem has more parts, but to get there I need to understand this one. Thank you.
[edit:] To clarify the aspects in Florian's comment: I don't understand why $-\Delta z_n$ has the given formula, and don't know how the measures $\mu_n,\nu_n$ are defined. In my intuition, $-\Delta z_n=0$ since this is the condition on $C_{ij}^n$ and outside this annulus $z_n$ is constant.