Paolo Musolino

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 connected open subset of such that 0 ∈ Ω and that is connected. Then we take q11, … ,qnn ∈ ]0,+ ∞ [and . If e is a small positive number, then we define the periodically perforated domain , where {e1, … ,en} is the canonical basis of . For e small and positive, we introduce a particular Dirichlet problem for the Laplace operator in the set . Namely, we consider a Dirichlet condition on the boundary of the set p + eΩ, 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 e and of the Dirichlet datum on p + e∂Ω, around a degenerate pair with e = 0. Copyright

Series expansions for the solution of the Dirichlet problem in a planar domain with a small hole

We consider the Dirichlet problem for the Laplace equation in a planar domain with a small hole. The diameter of the hole is proportional to a real parameter e and we denote by ue the corresponding solution. If p is a point of the domain, then for e small we write ue(p) as a convergent power series of e and of 1/(r0 + (2π) −1 log |e|), with r0 ∈ R. The coefficients of such series are given in terms of solutions of recursive systems of integral equations. We obtain a simplified expression for the series expansion of ue(p) in the case of a ring domain.

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

A singularly perturbed nonlinear Robin problem in a periodically perforated domain: a functional analytic approach†

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

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

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

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

around the degenerate value

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

\epsilon=0

A Functional Analytic Approach to Homogenization Problems

, 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

Let n ∈ ℕ∖{0, 1}. Let q be the n × n diagonal matrix with entries q 11, … , q nn in] 0, +∞[. Then qℤ n is a q-periodic lattice in ℝ n with fundamental cell . Let p ∈ Q. Let Ω be a bounded open subset of ℝ n containing 0. Let G be a (nonlinear) map from ∂Ω × ℝ to ℝ. Let γ be a positive-valued function defined on a right neighbourhood of 0 in the real line. Then we consider the problem for ε > 0 small, where ν p+εΩ denotes the outward unit normal to p + ε∂Ω. Under suitable assumptions and under condition limε→0+γ(ε)−1ε ∈ ℝ, we prove that 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.