Matteo Dalla Riva

Microscopically weakly singularly perturbed loads for a nonlinear traction boundary value problem: a functional analytic approach

Let Ω i and Ω o be two bounded open subsets of ℝ n containing 0. Let G i be a (nonlinear) map of ∂Ω i × ℝ n to ℝ n . Let a o be a map of ∂Ω o to the set M n (ℝ) of n × n matrices with real entries. Let g be a function of ∂Ω o to ℝ n . Let γ be a positive valued function defined on a right neighbourhood of 0 on the real line. Let T be a map of] 1 − (2/n), +∞[×M n (ℝ) to M n (ℝ). Then we consider the problem where νεΩ i and ν o denote the outward unit normal to ε∂Ω i and ∂Ω o , respectively, and where ε > 0 is a small parameter. Here (ω − 1) plays the role of ratio between the first and second Lamé constants and T(ω, ·) plays the role of (a constant multiple of) the linearized Piola Kirchhoff stress tensor. Under the condition that limε→0 γ−1(ε)ε(log ε)δ2,n exists in ℝ, we prove that under suitable assumptions the above problem has a family of solutions {u(ε, ·)}ε∈]0,ε′[ for ε′ sufficiently small and we analyse the behaviour of such a family as ε approaches 0 by an approach which is alternative to those of asymptotic analysis. Here δ2,n denotes the Kronecker symbol.

A Singularly Perturbed Nonideal 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

Stokes flow in a singularly perturbed exterior domain

\epsilon

A singularly perturbed nonlinear traction boundary value problem for linearized elastostatics. A functional analytic approach

. Under suitable assumptions, we show that the effective conductivity can be continued real analytically in the parameter

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

\epsilon

A Family of Fundamental Solutions of Elliptic Partial Differential Operators with Real Constant Coefficients

around the degenerate value

A Local Uniqueness Result for a Quasi-linear Heat Transmission Problem in a Periodic Two-phase Dilute Composite

\epsilon=0

Converging Expansions for Lipschitz Self-Similar Perforations of a Plane Sector

, in correspondence of which the inclusions collapse to points. Part of the results presented here have been announced in [M. Dalla Riva and P. Musolino, AIP Conf. Proc. 1493, American Institute of Physics, Melville, NY, 2012, pp. 264--268].

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

We consider a pair of domains Ω b and Ω s in ℝ n and we assume that the closure of Ω b does not intersect the closure of εΩ s for ε ∈] 0, ε0[. Then for a fixed ε ∈] 0, ε0 [we consider a boundary value problem in ℝ n ∖(Ω b  ∪ εΩ s ) which describes the steady state Stokes flow of an incompressible viscous fluid past a body occupying the domain Ω b and past a small impurity occupying the domain εΩ s . The unknowns of the problem are the velocity field u and the pressure field p, and we impose the value of the velocity field u on the boundary both of the body and of the impurity. We assume that the boundary velocity on the impurity displays an arbitrarily strong singularity when ε tends to 0. The goal is to understand the behaviour of (u, p) for ε small and positive. The methods developed aim at representing the limiting behaviour in terms of analytic maps and possibly singular but completely known functions of ε, such as ε−1, log ε.

Weakly Singular and Microscopically Hypersingular Load Perturbation for a Nonlinear Traction Boundary Value Problem: a Functional Analytic Approach

where ν i and ν o denote the outward unit normal to ∂ i and ∂ o, respectively, and where > 0 is a small parameter. Here (ω−1) plays the role of ratio between the first and second Lamé constants, and T(ω, ·) plays the role of (a constant multiple of) the linearized Piola Kirchhoff stress tensor, and Go plays the role of (a constant multiple of) a traction applied on the points of ∂ o. Then we prove that under suitable assumptions the above problem has a family of solutions {u( , ·)} ∈ ]0, ′[ for ′ sufficiently small and we show that in a certain sense {u( , ·)} ∈ ]0, ′[ can be continued real analytically for negative values of .