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Paolo Musolino

Publications and source records attributed to Paolo Musolino.

35 records · Page 2Linked to original sources
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.

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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 $ε=0$.

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arXiv2020

Shape perturbation of Grushin eigenvalues

We consider the spectral problem for the Grushin Laplacian subject to homogeneous Dirichlet boundary conditions on a bounded open subset of $\mathbb{R}^N$. We prove that the symmetric functions of the eigenvalues depend real analytically upon domain perturbations and we prove an Hadamard-type formula for their shape differential. In the case of perturbations depending on a single scalar parameter, we prove a Rellich-Nagy-type theorem which describes the bifurcation phenomenon of multiple eigenvalues. As corollaries, we characterize the critical shapes under isovolumetric and isoperimetric perturbations in terms of overdetermined problems and we deduce a new proof of the Rellich-Pohozaev identity for the Grushin eigenvalues.

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arXiv2019

Asymptotic behavior of $u$-capacities and singular perturbations for the Dirichlet-Laplacian

In this paper we study the asymptotic behavior of $u$-capacities of small sets and its application to the analysis of the eigenvalues of the Dirichlet-Laplacian on a bounded planar domain with a small hole. More precisely, we consider two (sufficiently regular) bounded open connected sets $Ω$ and $ω$ of $\mathbb{R}^2$, containing the origin. First, if $\varepsilon$ is positive and small enough and if $u$ is a function defined on $Ω$, we compute an asymptotic expansion of the $u$-capacity $\mathrm{Cap}_Ω(\varepsilon ω, u)$ as $\varepsilon \to 0$. As a byproduct, we compute an asymptotic expansion for the $N$-th eigenvalues of the Dirichlet-Laplacian in the perforated set $Ω\setminus (\varepsilon \overlineω)$ for $\varepsilon$ close to $0$. Such formula shows explicitly the dependence of the asymptotic expansion on the behavior of the corresponding eigenfunction near $0$ and on the shape $ω$ of the hole.

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arXiv2019

A mixed problem for the Laplace operator in a domain with moderately close holes

We investigate the behavior of the solution of a mixed problem in a domain with two moderately close holes. We introduce a positive parameter $ε$ and we define a perforated domain $Ω_ε$ obtained by making two small perforations in an open set. Both the size and the distance of the cavities tend to $0$ as $ε\to 0$. For $ε$ small, we denote by $u_ε$ the solution of a mixed problem for the Laplace equation in $Ω_ε$. We describe what happens to $u_ε$ as $ε\to 0$ in terms of real analytic maps and we compute an asymptotic expansion.

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arXiv2017

A Dirichlet problem for the Laplace operator in a domain with a small hole close to the boundary

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.

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arXiv2017

A Dirichlet problem for the Laplace operator in a domain with a small hole close to the boundary

We take an open regular domain $Ω$ in $\mathbb{R}^n$ with $n\ge 3$. We introduce a pair of positive parameters $ε_1$ and $ε_2$ and we set $ε\equiv(ε_1,ε_2)$. Then we define the perforated domain $Ω_ε$ by making in $Ω$ a small hole of size $ε_1ε_2$ at distance $ε_1$ from the boundary. When $ε\rightarrow(0, 0)$, the hole approaches the boundary while its size shrinks at a faster rate. In $Ω_ε$ we consider a Dirichlet problem for the Laplace equation and we denote its solution by $u_ε$. By an approach based on functional analysis and on the introduction of special layer potentials we show that the map which takes $ε$ to (a restriction of) $u_ε$ has a real analytic continuation in a neighbourhood of $(0, 0)$.

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arXiv2017

Mapping properties of weakly singular periodic volume potentials in Roumieu classes

The analysis of the dependence of integral operators on perturbations plays an important role in the study of inverse problems and of perturbed boundary value problems. In this paper we focus on the mapping properties of the volume potentials with weakly singular periodic kernels. Our main result is to prove that the map which takes a density function and a periodic kernel to a (suitable restriction of the) volume potential is bilinear and continuous with values in a Roumieu class of analytic functions. Such result extends to the periodic case some previous results obtained by the authors for non periodic potentials and it is motivated by the study of perturbation problems for the solutions of boundary value problems for elliptic differential equations in periodic domains.

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arXiv2017

Converging expansions for Lipschitz self-similar perforations of a plane sector

In contrast with the well-known methods of matching asymptotics and multiscale (or compound) asymptotics, the " functional analytic approach " of Lanza de Cristoforis (Analysis 28, 2008) allows to prove convergence of expansions around interior small holes of size $ε$ for solutions of elliptic boundary value problems. Using the method of layer potentials, the asymptotic behavior of the solution as $ε$ tends to zero is described not only by asymptotic series in powers of $ε$, but by convergent power series. Here we use this method to investigate the Dirichlet problem for the Laplace operator where holes are collapsing at a polygonal corner of opening $ω$. Then in addition to the scale $ε$ there appears the scale $η= ε^{π/ω}$. We prove that when $π/ω$ is irrational, the solution of the Dirichlet problem is given by convergent series in powers of these two small parameters. Due to interference of the two scales, this convergence is obtained, in full generality, by grouping together integer powers of the two scales that are very close to each other. Nevertheless, there exists a dense subset of openings $ω$ (characterized by Diophantine approximation properties), for which real analyticity in the two variables $ε$ and $η$ holds and the power series converge unconditionally. When $π/ω$ is rational, the series are unconditionally convergent, but contain terms in log $ε$.

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arXiv2013

A functional analytic approach for a singularly perturbed Dirichlet problem for the Laplace operator in a periodically perforated domain

We consider a sufficiently regular bounded open connected subset $Ω$ of $\mathbb{R}^n$ such that $0 \in Ω$ and such that $\mathbb{R}^n \setminus \clΩ$ is connected. Then we choose a point $w \in ]0,1[^n$. If $ε$ is a small positive real number, then we define the periodically perforated domain $T(ε) \equiv \mathbb{R}^n\setminus \cup_{z \in \mathbb{Z}^n}\cl(w+εΩ+z)$. For each small positive $ε$, we introduce a particular Dirichlet problem for the Laplace operator in the set $T(ε)$. More precisely, we consider a Dirichlet condition on the boundary of the set $w+εΩ$, and we denote the unique periodic solution of this problem by $u[ε]$. Then we show that (suitable restrictions of) $u[ε]$ can be continued real analytically in the parameter $ε$ around $ε=0$.

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arXiv2013

A singularly perturbed Dirichlet problem for the Poisson equation in a periodically perforated domain. A functional analytic approach

Let $Ω$ be a sufficiently regular bounded open connected subset of $\mathbb{R}^n$ such that $0 \in Ω$ and that $\mathbb{R}^n \setminus \mathrm{cl}Ω$ is connected. Then we take $(q_{11},\dots, q_{nn})\in ]0,+\infty[^n$ and $p \in Q\equiv \prod_{j=1}^{n}]0,q_{jj}[$. If $ε$ is a small positive number, then we define the periodically perforated domain $\mathbb{S}[Ω_{p,ε}]^{-} \equiv \mathbb{R}^n\setminus \cup_{z \in \mathbb{Z}^n}\mathrm{cl}\bigl(p+εΩ+\sum_{j=1}^n (q_{jj}z_j)e_j\bigr)$, where $\{e_1,\dots,e_n\}$ is the canonical basis of $\mathbb{R}^n$. For $ε$ small and positive, we introduce a particular Dirichlet problem for the Poisson equation in the set $\mathbb{S}[Ω_{p,ε}]^{-}$. Namely, we consider a Dirichlet condition on the boundary of the set $p+εΩ$, together with a periodicity condition. Then we show real analytic continuation properties of the solution as a function of $ε$, of the Dirichlet datum on $p+ε\partial Ω$, and of the Poisson datum, around a degenerate triple with $ε=0$.

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arXiv2013

A singularly perturbed Dirichlet problem for the Laplace operator in a periodically perforated domain. A functional analytic approach

Let $Ω$ be a sufficiently regular bounded open connected subset of $\mathbb{R}^n$ such that $0 \in Ω$ and that $\mathbb{R}^n \setminus \mathrm{cl}Ω$ is connected. Then we take $q_{11},..., q_{nn}\in ]0,+\infty[$ and $p \in Q\equiv \prod_{j=1}^{n}]0,q_{jj}[$. If $ε$ is a small positive number, then we define the periodically perforated domain $\mathbb{S}[Ω_ε]^{-} \equiv \mathbb{R}^n\setminus \cup_{z \in \mathbb{Z}^n}\mathrm{cl}\bigl(p+εΩ+\sum_{j=1}^n (q_{jj}z_j)e_j\bigr)$, where $\{e_1,...,e_n\}$ is the canonical basis of $\mathbb{R}^n$. For $ε$ small and positive, we introduce a particular Dirichlet problem for the Laplace operator in the set $\mathbb{S}[Ω_ε]^{-}$. Namely, we consider a Dirichlet condition on the boundary of the set $p+εΩ$, together with a periodicity condition. Then we show real analytic continuation properties of the solution and of the corresponding energy integral as functionals of the pair of $ε$ and of the Dirichlet datum on $p+ε\partial Ω$, around a degenerate pair with $ε=0$.

math.AP↗
arXiv2013

A singularly perturbed nonlinear traction problem in a periodically perforated domain. A functional analytic approach

We consider a periodically perforated domain obtained by making in R^n a periodic set of holes, each of them of size proportional to ε. Then we introduce a nonlinear boundary value problem for the Lamé equations in such a periodically perforated domain. The unknown of the problem is a vector valued function u which represents the displacement attained in the equilibrium configuration by the points of a periodic linearly elastic matrix with a hole of size εcontained in each periodic cell. We assume that the traction exerted by the matrix on the boundary of each hole depends (nonlinearly) on the displacement attained by the points of the boundary of the hole. Then our aim is to describe what happens to the displacement vector function u when εtends to 0. Under suitable assumptions we prove the existence of a family of solutions {u(ε,.) : ε\in ]0,ε'[} with a prescribed limiting behaviour when εapproaches 0. Moreover, the family {u(ε,.) : ε\in ]0,ε'[} is in a sense locally unique and can be continued real analytically for negative values of ε.

math.AP↗
arXiv2013

Real analytic families of harmonic functions in a domain with a small hole

Let n\ge 3. Let Ω^i and Ω^o be open bounded connected subsets of R^n containing the origin. Let ε_0>0 be such that Ω^o contains the closure of εΩ^i for all ε\in]-ε_0,ε_0[. Then, for a fixed ε\in]-ε_0,ε_0[\{0} we consider a Dirichlet problem for the Laplace operator in the perforated domain Ω^o\εΩ^i. We denote by u_εthe corresponding solution. If p\inΩ^o and p\neq 0, then we know that under suitable regularity assumptions there exist ε_p>0 and a real analytic operator U_p from ]-ε_p,ε_p[ to R such that u_ε(p)=U_p[ε] for all ε\in]0,ε_p[. Thus it is natural to ask what happens to the equality u_ε(p)=U_p[ε] for ε<0. We show a general result on continuation properties of some particular real analytic families of harmonic functions in domains with a small hole and we prove that the validity of the equality u_ε(p)=U_p[ε] for ε<0 depends on the parity of the dimension n.

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arXiv2013

Effective conductivity of a singularly perturbed periodic two-phase composite with imperfect thermal contact at the two-phase interface

We consider the asymptotic behaviour of the effective thermal conductivity of a two-phase composite obtained by introducing into an infinite homogeneous matrix a periodic set of inclusions of a different material and of size proportional to a positive parameter ε. We are interested in the case of imperfect thermal contact at the two-phase interface. Under suitable assumptions, we show that the effective thermal conductivity can be continued real analytically in the parameter εaround the degenerate value ε=0, in correspondence of which the inclusions collapse to points. The results presented here are obtained by means of an approach based on functional analysis and potential theory and are also part of a forthcoming paper by the authors.

math.AP↗
arXiv2013

A singularly perturbed non-ideal transmission problem and application to the effective conductivity of a periodic composite

We investigate the effective thermal conductivity of a two-phase composite with thermal resistance at the interface. The composite is obtained by introducing into an infinite homogeneous matrix a periodic set of inclusions of a different material. The diameter of each inclusion is assumed to be proportional to a positive real parameter ε. Under suitable assumptions, we show that the effective conductivity can be continued real analytically in the parameter εaround the degenerate value ε=0, in correspondence of which the inclusions collapse to points.

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