arXiv subjects
Publications and source records attributed to Paolo Musolino.
We fix the proof of Theorem 1.17 in the quoted reference, which does not in general gives the correct coefficients in the asymptotic behavior of eigenvalues for the Dirichlet Laplacian when a small hole is removed from the domain.
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.
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.
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.
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.
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.
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.
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.
We provide a full series expansion of a generalization of the so-called $u$-capacity related to the Dirichlet-Laplacian in dimension three and higher, extending previous results of the authors, and of the authors together with Virginie Bonnaillie-Noël, dealing with the planar case. We apply the result in order to study the asymptotic behavior of perturbed eigenvalues when Dirichlet conditions are imposed on a small regular subset of the domain of the eigenvalue problem.
We consider a perforated domain $Ω(ε)$ of $\mathbb{R}^2$ with a small hole of size $ε$ and we study the behavior of the solution of a mixed Neumann-Robin problem in $Ω(ε)$ as the size $ε$ of the small hole tends to $0$. In addition to the geometric degeneracy of the problem, the $ε$-dependent Robin condition may degenerate into a Neumann condition for $ε=0$ and the Robin datum may diverge to infinity. Our goal is to analyze the asymptotic behavior of the solutions to the problem as $ε$ tends to $0$ and understand how the boundary condition affects the behavior of the solutions when $ε$ is close to $0$.
We consider the Laplace equation in a domain of $\mathbb{R}^n$, $n\ge 3$, with a small inclusion of size $ε$. On the boundary of the inclusion we define a nonlinear nonautonomous transmission condition. For $ε$ small enough one can prove that the problem has solutions. In this paper, we study the local uniqueness of such solutions.
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 $ϕ$.
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 $f/2π$.
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.
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.
Taking advantage from the so-called "Lemma on small eigenvalues" by Colin de Verdière, we study ramification for multiple eigenvalues of the Dirichlet Laplacian in bounded perforated domains. The asymptotic behavior of multiple eigenvalues turns out to depend on the asymptotic expansion of suitable associated eigenfunctions. We treat the case of planar domains in details, thanks to the asymptotic expansion of a generalization of the so-called u-capacity which we compute in dimension 2. In this case multiple eigenvalues are proved to split essentially by different rates of convergence of the perturbed eigenvalues or by different coefficients in front of their expansion if the rate of two eigenbranches turns out to be the same.
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.
We study the asymptotic behavior of the solutions of a boundary value problem for the Laplace equation in a perforated domain in $\mathbb{R}^n$, $n\geq 3$, with a (nonlinear) Robin boundary condition on the boundary of the small hole. The problem we wish to consider degenerates under three aspects: in the limit case the Robin boundary condition may degenerate into a Neumann boundary condition, the Robin datum may tend to infinity, and the size $ε$ of the small hole where we consider the Robin condition collapses to $0$. We study how these three singularities interact and affect the asymptotic behavior as $ε$ tends to $0$, and we represent the solution and its energy integral in terms of real analytic maps and known functions of the singular perturbation parameters.