arXiv subjects
Publications and source records attributed to Matteo Dalla Riva.
We consider a Neumann problem for the Laplace equation in a periodic domain. We prove that the solution depends real analytically on the shape of the domain, on the periodicity parameters, on the Neumann datum, and on its boundary integral.
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.
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.
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 $(0, 0)$.
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.
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 $ε$.
We consider the spectral problem \begin{equation*} \left\{\begin{array}{ll} -Δu_{\varepsilon}=λ(\varepsilon)ρ_{\varepsilon}u_{\varepsilon} & {\rm in}\ Ω\\ \frac{\partial u_{\varepsilon}}{\partialν}=0 & {\rm on}\ \partialΩ\end{array}\right. \end{equation*} in a smooth bounded domain $Ω$ of $\mathbb R^2$. The factor $ρ_{\varepsilon}$ which appears in the first equation plays the role of a mass density and it is equal to a constant of order $\varepsilon^{-1}$ in an $\varepsilon$-neighborhood of the boundary and to a constant of order $\varepsilon$ in the rest of $Ω$. We study the asymptotic behavior of the eigenvalues $λ(\varepsilon)$ and the eigenfunctions $u_{\varepsilon}$ as $\varepsilon$ 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.
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.
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.
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.
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.
We construct of a family of fundamental solutions for elliptic partial differential operators with real constant coefficients. The elements of such a family are expressed by means of jointly real analytic functions of the coefficients of the operators and of the spatial variable. We show regularity properties in the frame of Schauder spaces for the corresponding single layer potentials.