Preamble
In a recent problem that I have found is about the limit of weighted delta functions in two dimensions.
Background
Let $\Omega \subset \mathbb{R}^2$ be bounded with sufficently smooth boundary, constant $\lambda_{\Omega}>0$ and order parameter $\epsilon \ll 1$. Let us assume that the sequence $\{x^i_\epsilon\}_{i=1}^{N_{\epsilon}}\subset \Omega$ satisifies the following
- The seqeunce is not too close to the boundary: $dist(x^i_{\epsilon},\partial \Omega) \geqslant \lambda_{\Omega}\epsilon$ for all $i = 1, \ldots, N_{\epsilon}$.
- The points are $\epsilon$ separated: $\inf\limits_{i \neq j}|x^i_{\epsilon}-x^j_{\epsilon}| \geqslant 2 \lambda_{\Omega}\epsilon$.
- The points are approximately uniformly distributed and do not cluster: $$\epsilon^2 \sum\limits_{i=1}^{N_{\epsilon}}\delta(\cdot-x^i_{\epsilon}) \rightharpoonup^* dx\mathcal{L}\Omega$$ where $dx\mathcal{L}\Omega$ is the lebesgue measure in $\Omega$.
Remark
The first two conditions imply that $N_{\epsilon}\sim \frac{1}{\epsilon^2}$.
Problem set-up
For a given constants $d \in \mathbb{Z}$ and $C \in \mathbb{N}_0$, independent of $\epsilon$ we consider a sequence of integers $d^i_{\epsilon} \in \mathbb{Z}$ for $i=1,\ldots, N_{\epsilon}$ which satisfy:
- They are bounded: $|d^i_{\epsilon}|\leqslant C$.
- Their summation is constant for all $\epsilon$: $$d = \sum\limits_{i=1}^{N_{\epsilon}}d^i_{\epsilon}.$$ We now define the weighted discrete measure $\mu_{\epsilon}$ by $$\mu_{\epsilon}:= \epsilon^2 \sum\limits_{i=1}^{N_{\epsilon}}d^i_{\epsilon} \delta(\cdot-x^i_{\epsilon}).$$ We wish to investigate the behaviour of $\mu_{\epsilon}$ as $\epsilon \rightarrow 0$.
Problem
We know by definition we have that $$\int_{\Omega} \psi \mu_{\epsilon} = \epsilon^2\sum\limits_{i=1}^{N_{\epsilon}}d^i_{\epsilon}\psi(x^i_\epsilon). \quad \forall \psi \in C^{\infty}(\Omega).$$ This appears similar to a Riemann integral, with additional terms, thus I would expect that $$\int_{\Omega} \psi \mu_{\epsilon} \rightharpoonup^* \int\limits_{\Omega}\rho(x)\psi(x)dx\mathcal{L}\Omega$$ for some density function $\rho$ which is dependent on the sequence $d^i_{\epsilon}$. My questions are the following:
- Can $\rho$ be explicitally given?
- If not what additional conditions are necessary on $d^i_{\epsilon}$ for $\rho$ to be given explicitly?
- Can anything be said about the limiting behaviour of $\epsilon^{-2}\mu_{\epsilon}$?