Matteo Dalla Riva

Virginie Bonnaillie-Noël, Matteo Dalla Riva, Marc Dambrine et al.

We study the Dirichlet problem in a domain with a small hole close to the boundary. To do so, for each pair $\boldsymbol\varepsilon = (\varepsilon_1, \varepsilon_2 )$ of positive parameters, we consider a perforated domain $Ω_{\boldsymbol\varepsilon}$ obtained by making a small hole of size $\varepsilon_1 \varepsilon_2 $ in an open regular subset $Ω$ of $\mathbb{R}^n$ at distance $\varepsilon_1$ from the boundary $\partialΩ$. As $\varepsilon_1 \to 0$, the perforation shrinks to a point and, at the same time, approaches the boundary. When $\boldsymbol\varepsilon \to (0,0)$, the size of the hole shrinks at a faster rate than its approach to the boundary. We denote by $u_{\boldsymbol\varepsilon}$ the solution of a Dirichlet problem for the Laplace equation in $Ω_{\boldsymbol\varepsilon}$. For a space dimension $n\geq 3$, we show that the function mapping $\boldsymbol\varepsilon$ to $u_{\boldsymbol\varepsilon}$ has a real analytic continuation in a neighborhood of $(0,0)$. By contrast, for $n=2$ we consider two different regimes: $\boldsymbol\varepsilon$ tends to $(0,0)$, and $\varepsilon_1$ tends to $0$ with $\varepsilon_2$ fixed. When $\boldsymbol\varepsilon\to(0,0)$, the solution $u_{\boldsymbol\varepsilon}$ has a logarithmic behavior; when only $\varepsilon_1\to0$ and $\varepsilon_2$ is fixed, the asymptotic behavior of the solution can be described in terms of real analytic functions of $\varepsilon_1$. We also show that for $n=2$, the energy integral and the total flux on the exterior boundary have different limiting values in the two regimes. We prove these results by using functional analysis methods in conjunction with certain special layer potentials.

Giulio Valentino Dalla Riva, Matteo Dalla Riva

Latent-position random graph models usually treat the node set as fixed once the sample size is chosen, while graphon-based and random-measure constructions allow more randomness at the cost of weaker geometric interpretability. We introduce \emph{Intensity Dot Product Graphs} (IDPGs), which extend Random Dot Product Graphs by replacing a fixed collection of latent positions with a Poisson point process on a Euclidean latent space. This yields a model with random node populations, RDPG-style dot-product affinities, and a population-level intensity that links continuous latent structure to finite observed graphs. We define the heat map and the desire operator as continuous analogues of the probability matrix, prove a spectral consistency result connecting adjacency singular values to the operator spectrum, compare the construction with graphon and digraphon representations, and show how classical RDPGs arise in a concentrated limit. Because the model is parameterized by an evolving intensity, temporal extensions through partial differential equations arise naturally.