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Matteo Dalla Riva

Publications and source records attributed to Matteo Dalla Riva.

At least 19 recordsLinked to original sources
arXiv2026

Intensity Dot Product Graphs

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.

stat.ML↗
arXiv2026

Regularity properties of certain convolution operators in Hölder spaces

The aim of this paper is to prove a theorem of C.~Miranda on the Hölder regularity of convolution operators acting on the boundary of an open set in the limiting case in which the open set is of class $C^{1,1}$ and the densities are of class $C^{0,1}$. The convolution operators that we consider are generalizations of those that are associated to layer potential operators, which are a useful tool for the analysis of boundary value problems.

math.AP↗
arXiv2025

Shape sensitivity analysis of the heat equation and the Dirichlet-to-Neumann map

We study a Dirichlet problem for the heat equation in a domain containing an interior hole. The domain has a fixed outer boundary and a variable inner boundary determined by a diffeomorphism $ϕ$. We analyze the maps that assign to the infinite-dimensional shape parameter $ϕ$ the corresponding solution and its normal derivative, and we prove that both are smooth. Motivated by an application to an inverse problem, we then compute the differential with respect to $ϕ$ of the normal derivative of the solution on the exterior boundary.

math.AP↗
arXiv2025

Periodic layer potentials and domain perturbations

In this paper, we review the construction of periodic fundamental solutions and periodic layer potentials for various differential operators. Specifically, we focus on the Laplace equation, the Helmholtz equation, the Lamé system, and the heat equation. We then describe how these layer potentials can be applied to analyze domain perturbation problems. In particular, we present applications to the asymptotic behavior of quasi-periodic solutions for a Dirichlet problem for the Helmholtz equation in an unbounded domain with small periodic perforations. Additionally, we investigate the dependence of spatially periodic solutions of an initial value Dirichlet problem for the heat equation on regular perturbations of the base of a parabolic cylinder.

math.AP↗
arXiv2025

Shape sensitivity analysis of Neumann-Poincaré eigenvalues

This paper concerns the eigenvalues of the Neumann-Poincaré operator, a boundary integral operator associated with the harmonic double-layer potential. Specifically, we examine how the eigenvalues depend on the support of integration and prove that the map associating the support's shape to the eigenvalues is real-analytic. We then compute its first derivative and present applications of the resulting formula. The proposed method allows for handling infinite-dimensional perturbation parameters for multiple eigenvalues and perturbations that are not necessarily in the normal direction.

math.AP↗
arXiv2024

Shape perturbation of a nonlinear mixed problem for the heat equation

We consider the heat equation in a domain that has a hole in its interior. We impose a Neumann condition on the exterior boundary and a nonlinear Robin condition on the boundary of the hole. The shape of the hole is determined by a suitable diffeomorphism $ϕ$ defined on the boundary of a reference domain. Assuming that the problem has a solution $u_0$ when $ϕ$ is the identity map, we demonstrate that a solution $u_ϕ$ continues to exist for $ϕ$ close to the identity map and that the "domain-to-solution" map $ϕ\mapsto u_ϕ$ is of class $C^\infty$. Moreover, we show that the family of solutions $\{u_ϕ\}_ϕ$ is, in a sense, locally unique. Our argument relies on tools from Potential Theory and the Implicit Function Theorem. Some remarks a the linear case complete the paper.

math.AP↗
arXiv2024

Dirichlet problem on perturbed conical domains via converging generalized power series

We consider the Poisson equation with homogeneous Dirichlet conditions in a family of domains in $R^{n}$ indexed by a small parameter $ε$. The domains depend on $ε$ only within a ball of radius proportional to $ε$ and, as $ε$ tends to zero, they converge in a self-similar way to a domain with a conical boundary singularity. We construct an expansion of the solution as a series of fractional powers of $ε$, and prove that it is not just an asymptotic expansion as $ε\to0$, but that, for small values of $ε$, it converges normally in the Sobolev space $H^{1}$. The phenomenon that solutions to boundary value problems on singularly perturbed domains may have convergent expansions is the subject of the Functional Analytic Approach by Lanza de Cristoforis and his collaborators. This approach was originally adopted to study small holes shrinking to interior points of a smooth domain and heavily relies on integral representations obtained through layer potentials. To relax all regularity assumptions, we forgo boundary layer potentials and instead exploit expansions in terms of eigenfunctions of the Laplace-Beltrami operator on the intersection of the cone with the unit sphere. Our analysis is based on a two-scale cross-cutoff ansatz for the solution. Specifically, we write the solution as a sum of a function in the slow variable multiplied by a cutoff function depending on the fast variable, plus a function in the fast variable multiplied by a cutoff function depending on the slow variable. While the cutoffs are considered fixed, the two unknown functions are solutions to a $2\times2$ system of partial differential equations that depend on $ε$ in a way that can be analyzed in the framework of generalized power series when the right-hand side of the Poisson equation vanishes in a neighborhood of the perturbation.

math.AP↗
arXiv2023

Multi-parameter perturbations for the space-periodic heat equation

This paper is divided into three parts. The first part focuses on periodic layer heat potentials, demonstrating their smooth dependence on regular perturbations of the support of integration. In the second part, we present an application of the results from the first part. Specifically, we consider a transmission problem for the heat equation in a periodic two-phase composite material and we show that the solution depends smoothly on the shape of the transmission interface, boundary data, and conductivity parameters. Finally, in the last part of the paper, we fix all parameters except for the contrast parameter and outline a strategy to deduce an explicit expansion of the solution using a Neumann-type series.

math.AP↗
arXiv2023

Continuous harmonic functions on a ball that are not in $H^s$ for $s>1/2$

We show that there are harmonic functions on a ball ${\mathbb{B}_n}$ of $\mathbb{R}^n$, $n\ge 2$, that are continuous up to the boundary (and even Hölder continuous) but not in the Sobolev space $H^s(\mathbb{B}_n)$ for any $s$ sufficiently big. The idea for the construction of these functions is inspired by the two-dimensional example of a harmonic continuous function with infinite energy presented by Hadamard in 1906. To obtain examples in any dimension $n\ge 2$ we exploit certain series of spherical harmonics. As an application, we verify that the regularity of the solutions that was proven for a class of boundary value problems with nonlinear transmission conditions is, in a sense, optimal.

math.AP↗
arXiv2022

Existence results for a nonlinear nonautonomus transmission problem via domain perturbation

In this paper we study the existence and the analytic dependence upon domain perturbation of the solutions of a nonlinear nonautonomous transmission problem for the Laplace equation. The problem is defined in a pair of sets consisting of a perforated domain and an inclusion whose shape is determined by a suitable diffeomorphism $ϕ$. First we analyse the case in which the inclusion is a fixed domain. Then we will perturb the inclusion and study the arising boundary value problem and the dependence of a specific family of solutions upon the perturbation parameter $ϕ$.

math.AP↗
arXiv2022

Singular behavior for a multi-parameter periodic Dirichlet problem

We consider a Dirichlet problem for the Poisson equation in a periodically perforated domain. The geometry of the domain is controlled by two parameters: a real number $ε>0$ proportional to the radius of the holes and a map $ϕ$, which models the shape of the holes. So, if $g$ denotes the Dirichlet boundary datum and $f$ the Poisson datum, we have a solution for each quadruple $(ε,ϕ,g,f)$. Our aim is to study how the solution depends on $(ε,ϕ,g,f)$, especially when $ε$ is very small and the holes narrow to points. In contrast with previous works, we don't introduce the assumption that $f$ has zero integral on the fundamental periodicity cell. This brings in a certain singular behavior for $ε$ close to $0$. We show that, when the dimension $n$ of the ambient space is greater than or equal to $3$, a suitable restriction of the solution can be represented with an analytic map of the quadruple $(ε,ϕ,g,f)$ multiplied by the factor $1/ε^{n-2}$. In case of dimension $n=2$, we have to add $\log ε$ times the integral of $f/2π$.

math.AP↗
arXiv2022

The Functional Analytic Approach for quasi-periodic boundary value problems for the Helmholtz equation

We lay down the preliminary work to apply the Functional Analytic Approach to quasi-periodic boundary value problems for the Helmholtz equation. This consists in introducing a quasi-periodic fundamental solution and the related layer potentials, showing how they are used to construct the solutions of quasi-periodic boundary value problems, and how they behave when we perform a singular perturbation of the domain. To show an application, we study a nonlinear quasi-periodic Robin problem in a domain with a set of holes that shrink to points.

math.AP↗
arXiv2022

A degenerating Robin-type traction problem in a periodic domain

We consider a linearly elastic material with a periodic set of voids. On the boundaries of the voids we set a Robin-type traction condition. Then we investigate the asymptotic behavior of the displacement solution as the Robin condition turns into a pure traction one. To wit, there will be a matrix function {$b[k](\cdot)$ that depends analytically on a real parameter $k$ and vanishes for $k=0$ and we multiply the Dirichlet-like part of the Robin condition by $b[k](\cdot)$}. We show that the displacement solution can be written in terms of power series of $k$ that converge for $k$ in a whole neighborhood of $0$. For our analysis we use the Functional Analytic Approach.

math.AP↗
arXiv2022

Integral equation method for a Robin-type traction problem in a periodic domain

In this note, we consider a Robin-type traction problem for a linearly elastic body occupying an infinite periodically perforated domain. After proving the uniqueness of the solution we use periodic elastic layer potentials to show that the solution can be written as the sum of a single layer potential, a constant function and a linear function of the space variable. The density of the periodic single layer potential and the constant are identified as the unique solutions of a certain integral equation.

math.AP↗
arXiv2022

Multi-parameter analysis of the obstacle scattering problem

We consider the acoustic field scattered by a bounded impenetrable obstacle and we study its dependence upon a certain set of parameters. As usual, the problem is modeled by an exterior Dirichlet problem for the Helmholtz equation $Δu +k^2u=0$. We show that the solution $u$ and its far field pattern $u_\infty$ depend real analytically on the shape of the obstacle, the wave number $k$, and the Dirichlet datum. We also prove a similar result for the corresponding Dirichlet-to-Neumann map.

math.AP↗
arXiv2022

Shape analyticity and singular perturbations for layer potential operators

We study the effect of regular and singular domain perturbations on layer potential operators for the Laplace equation. First, we consider layer potentials supported on a diffeomorphic image $ϕ(\partialΩ)$ of a reference set $\partialΩ$ and we present some real analyticity results for the dependence upon the map $ϕ$. Then we introduce a perforated domain $Ω(ε)$ with a small hole of size $ε$ and we compute power series expansions that describe the layer potentials on $\partialΩ(ε)$ when the parameter $ε$ approximates the degenerate value $ε=0$.

math.AP↗