Papers by Matteo Dalla Riva
arXiv (Cornell University), Jun 26, 2013
Let n ≥ 3. Let Ω i and Ω o be open bounded connected subsets of R n containing the origin. Let ǫ ... more Let n ≥ 3. Let Ω i and Ω o be open bounded connected subsets of R n containing the origin. Let ǫ 0 > 0 be such that Ω o contains the closure of ǫΩ i for all ǫ ∈]ǫ 0 , ǫ 0 [. Then, for a fixed ǫ ∈]ǫ 0 , ǫ 0 [\{0} we consider a Dirichlet problem for the Laplace operator in the perforated domain Ω o \ ǫΩ i . We denote by u ǫ the corresponding solution. If p ∈ Ω o and p = 0, then we know that under suitable regularity assumptions there exist ǫ p > 0 and a real analytic operator Thus it is natural to ask what happens to the equality u ǫ (p) = U p [ǫ] for ǫ negative. 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 ǫ (p) = U p [ǫ] for ǫ negative depends on the parity of the dimension n.
arXiv (Cornell University), Aug 22, 2014
Let Ω o and Ω i be open bounded subsets of R n of class C 1,α such that the closure of Ω i is con... more Let Ω o and Ω i be open bounded subsets of R n of class C 1,α such that the closure of Ω i is contained in Ω o . Let f o be a function in C 1,α (∂Ω o ) and let F and G be continuous functions from ∂Ω i × R to R. By exploiting an argument based on potential theory and on the Leray-Schauder principle we show that under suitable and completely explicit conditions on F and G there exists at least one pair of continuous functions (u o , u i ) such that where the last equality is attained in certain weak sense. In a simple example we show that such a pair of functions (u o , u i ) is in general neither unique nor local unique. If instead the fourth condition of the problem is obtained by a small nonlinear perturbation of a homogeneous linear condition, then we can prove the existence of at least one classical solution which is in addition locally unique.
arXiv (Cornell University), May 5, 2017
We consider the spectral problem on ∂Ω in a smooth bounded domain Ω of R 2 . The factor ρ ε which... more We consider the spectral problem on ∂Ω in a smooth bounded domain Ω of R 2 . The factor ρ ε which appears in the first equation plays the role of a mass density and it is equal to a constant of order ε -1 in an ε-neighborhood of the boundary and to a constant of order ε in the rest of Ω. We study the asymptotic behavior of the eigenvalues λ(ε) and the eigenfunctions u ε as ε tends to zero. We obtain explicit formulas for the first and second terms of the corresponding asymptotic expansions by exploiting the solutions of certain auxiliary boundary value problems.
arXiv (Cornell University), May 5, 2017
We investigate a Dirichlet problem for the Laplace equation in a domain of R 2 with two small clo... more We investigate a Dirichlet problem for the Laplace equation in a domain of R 2 with two small close holes. The domain is obtained by making in a bounded open set two perforations at distance |ǫ 1 | one from the other and each one of size |ǫ 1 ǫ 2 |. In such a domain, we introduce a Dirichlet problem and we denote by u ǫ 1 ,ǫ 2 its solution. We show that the dependence of u ǫ 1 ,ǫ 2 upon (ǫ 1 , ǫ 2 ) can be described in terms of real analytic maps of the pair (ǫ 1 , ǫ 2 ) defined in an open neighborhood of (0, 0) and of logarithmic functions of ǫ 1 and ǫ 2 . Then we study the asymptotic behaviour of of u ǫ 1 ,ǫ 2 as ǫ 1 and ǫ 2 tend to zero. We show that the first two terms of an asymptotic approximation can be computed only if we introduce a suitable relation between ǫ 1 and ǫ 2 .
Mathematical Methods in The Applied Sciences, Dec 6, 2012
Integral Equations and Operator Theory, May 18, 2017
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
DOAJ (DOAJ: Directory of Open Access Journals), Nov 1, 2014
We present a limiting property and a local uniqueness result for converging families of solutions... more We present a limiting property and a local uniqueness result for converging families of solutions of a singularly perturbed nonlinear traction problem in an unbounded periodic domain with small holes.
Mathematical Methods in The Applied Sciences, May 28, 2013
We consider a periodically perforated domain obtained by making in R n a periodic set of holes, e... more We consider a periodically perforated domain obtained by making in R n a periodic set of holes, each of them of size proportional to ǫ. 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 ǫ contained 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 ǫ tends to 0. Under suitable assumptions we prove the existence of a family of solutions {u(ǫ, •)} ǫ∈]0,ǫ ′ [ with a prescribed limiting behaviour when ǫ approaches 0. Moreover, the family {u(ǫ, •)} ǫ∈]0,ǫ ′ [ is in a sense locally unique and can be continued real analytically for negative values of ǫ.
arXiv (Cornell University), Nov 23, 2022
We consider the Laplace equation in a domain of R n , n ≥ 3, with a small inclusion of size ǫ. On... more We consider the Laplace equation in a domain of R n , n ≥ 3, with a small inclusion of size ǫ. On the boundary of the inclusion we define a nonlinear nonautonomous transmission condition. For ǫ small enough one can prove that the problem has solutions. In this paper, we study the local uniqueness of such solutions.
arXiv (Cornell University), Mar 3, 2022
We study the effect of regular and singular domain perturbations on layer potential operators for... more 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(∂Ω) of a reference set ∂Ω and we present some real analyticity results for the dependence upon the map φ. Then we introduce a perforated domain Ω( ) with a small hole of size and we compute power series expansions that describe the layer potentials on ∂Ω( ) when the parameter approximates the degenerate value = 0.
arXiv (Cornell University), Mar 9, 2022
We show that there are harmonic functions on a ball Bn of R n , n ≥ 2, that are continuous, and e... more We show that there are harmonic functions on a ball Bn of R n , n ≥ 2, that are continuous, and even Hölder continuous, up to the boundary but not in the Sobolev space H s (Bn) for s bigger than a certain sharp bound. The idea for the construction is inspired by the two-dimensional example of a harmonic continuous function with infinite energy presented by To obtain examples in any dimension n ≥ 2 we exploit certain series of spherical harmonics.
Mathematical Modelling and Analysis
We consider a linearly elastic material with a periodic set of voids. On the boundaries of the vo... more 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](·) 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](·). 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.
arXiv (Cornell University), Oct 28, 2022
We lay down the preliminary work to apply the Functional Analytic Approach to quasi-periodic boun... more 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.
arXiv (Cornell University), Sep 6, 2022
We consider a linearly elastic material with a periodic set of voids. On the boundaries of the vo... more 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](•) 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](•). 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.
arXiv (Cornell University), Aug 4, 2022
In this note, we consider a Robin-type traction problem for a linearly elastic body occupying an ... more 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.
ESAIM: Mathematical Modelling and Numerical Analysis
We study the effect of regular and singular domain perturbations on layer potential operators for... more 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 imageϕ(∂Ω) of a reference set ∂Ω and we present some real analyticity results for the dependence upon the mapϕ. Then we introduce a perforated domain Ω(ε) with a small hole of sizeεand we compute power series expansions that describe the layer potentials on ∂Ω(ε) when the parameterεapproximates the degenerate valueε = 0.
arXiv (Cornell University), Mar 9, 2022
We show that there are harmonic functions on a ball Bn of R n , n ≥ 2, that are continuous, and e... more We show that there are harmonic functions on a ball Bn of R n , n ≥ 2, that are continuous, and even Hölder continuous, up to the boundary but not in the Sobolev space H s (Bn) for s bigger than a certain sharp bound. The idea for the construction is inspired by the two-dimensional example of a harmonic continuous function with infinite energy presented by To obtain examples in any dimension n ≥ 2 we exploit certain series of spherical harmonics.
Inverse Problems
We consider the acoustic field scattered by a bounded impenetrable obstacle and we study its depe... more 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 2 u = 0. We show that the solution u and its far field pattern u ∞ 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.
Mechanics and Physics of Structured Media, 2022
In this survey, we present some results on the behavior of effective properties in presence of pe... more In this survey, we present some results on the behavior of effective properties in presence of perturbations of the geometric and physical parameters. We first consider the case of a Newtonian fluid flowing at low Reynolds numbers around a periodic array of cylinders. We show the results of , where it is proven that the average longitudinal flow depends real analytically upon perturbations of the periodicity structure and the cross section of the cylinders. Next, we turn to the effective conductivity of a periodic two-phase composite with ideal contact at the interface. The composite is obtained by introducing a periodic set of inclusions into an infinite homogeneous matrix made of a different material. We show a result of [41] on the real analytic dependence of the effective conductivity upon perturbations of the shape of the inclusions, the periodicity structure, and the conductivity of each material. In the last part of the paper, we extend the result of to the case of a periodic two-phase composite with imperfect contact at the interface.
Journal of Integral Equations and Applications, 2020
The analysis of the dependence of integral operators on perturbations plays an important role in ... more 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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Papers by Matteo Dalla Riva