Umberto De Maio

• University of Naples Federico II

We investigate a degenerating elliptic problem in a multi-structure \Omega_\varepsilon of \mathbb{R}^3 , in the framework of the thermal stationary conduction with highly contrasting diffusivity. Precisely, \Omega_\varepsilon consists of a fixed basis \Omega^- surmounted by a thin cylinder \Omega_\varepsilon^+ with height 1 and cross-section with a...

We consider a Ginzburg-Landau type equation in $\R^2$ of the form $-\Delta u = u J'(1-|u|^{2})$ with a potential function $J$ satisfying weak conditions allowing for example a zero of infinite order in the origin. We extend in this context the results concerning quantization of finite potential solutions of H.Brezis, F.Merle, T.Rivi\`ere from \cite...

The paper studies a problem of an optimal control in coefficients for the system of two coupled elliptic equations also known as thermistor problem which provides a simultaneous description of the electric field \(u=u(x)\) and temperature \(\theta (x)\). The coefficient b of operator \(\mathrm {div}\,\left( b(x)\, \nabla \, \theta (x)\right) \) is...

We consider non-autonomous evolution inclusions and hemivariation inequalities with possibly non-monotone multidimensional “reaction-velocity” law. The dynamics of all weak solutions defined on the positive semi-axis of time is investigated. We prove the existence global attractor. New properties of complete trajectories are justified. The pointwis...

The main purpose is the study of optimal control problem in a domain with rough boundary for the mixed Dirichlet‐Neumann boundary value problem for the strongly nonlinear elliptic equation with exponential nonlinearity. A density of surface traction u acting on a part of rough boundary is taken as a control. The optimal control problem is to minimi...

In the paper we study boundary–value and spectral problems for the Laplacian operator in a domain with a smooth boundary. It is assumed that on a small part of the boundary there is a Dirichlet boundary condition and on all the rest boundary there is a Steklov condition. We study the behavior of the initial problem when a small parameter defining t...

We study an optimal control problem for the mixed Dirichlet–Neumann boundary value problem for the strongly non-linear elliptic equation with exponential nonlinearity in a domain with rugous boundary. A density of surface traction u acting on a part of rugous boundary is taken as a control. The optimal control problem is to minimize the discrepancy...

In the paper, we study the Steklov-type problem for the system of elasticity with rapidly changing boundary condition depending on a small parameter . We construct the homogenized (limit) problem and estimate the rate of convergence of the eigenvalues and eigenfunctions of the initial problem as .

We study the asymptotic behavior of solutions and eigenelements to a 2-dimensional and 3-dimensional boundary value problem for the Laplace equation in a domain perforated along part of the boundary. On the boundary of holes we set the homogeneous Dirichlet boundary condition and the Steklov spectral condition on the mentioned part of the outer bou...

We study the asymptotic behavior of solutions and eigenelements to a boundary value problem for the Laplace equation in a domain perforated along part of the boundary. On the boundary of holes, we set the homogeneous Dirichlet boundary condition and the Steklov spectral condition on the mentioned part of the outer boundary of the domain. Assuming t...

In this paper we study the asymptotic behavior of the solutions of time dependent micromagnetism problem in a multi-domain consisting of two joined ferromagnetic thin films. We distinguish different regimes depending on the limit of the ratio between the small thickness of the two films.

In this paper, we study the exact controllability of a second order linear evolution equation in a domain with highly oscillating boundary with homogeneous Neumann boundary condition on the oscillating part of boundary. Our aim is to obtain the exact controllability for the homogenized equation. The limit problem with Neumann condition on the oscil...

This paper is concerned with the study of homogenization of an optimal control problem governed by a second-order linear evolution equation with a homogeneous Neumann boundary condition in a domain bounded at the bottom by a smooth wall and at the top by a rough wall. The latter is assumed to consist in a plane wall covered with periodically distri...

In this paper, using Pontryagin’s maximum principle, we study the asymptotic behaviour of a parabolic optimal control problem in a domain \(\Omega _{\varepsilon }\subset \mathbf {R}^{n},\) whose boundary \(\partial \Omega _{\varepsilon }\) contains a highly oscillating part. On this part we consider a homogeneous Neumann boundary condition. We iden...

In this paper we study the asymptotic behavior of a quasy-stationary ferromagnetic problem in a multi-domain consisting of two joined thin films. It is possible to distinguish different regimes depending on the limit \(q\) of the ratio between the small thickness of the two films. Here the case \(q=0\) and \(q=+\infty \) are analyzed.

In the paper we deal with the homogenization problem for the Poisson equation in a singularly perturbed domain with multilevel oscillating boundary. This domain consists of the body, a large number of thin periodically situated cylinders joining to the body through thin random transmission zone with rapidly oscillating boundary. Inhomogeneous Fouri...

The main purpose of this paper is to derive a wall law for a flow over a very rough surface. We consider a viscous incompressible fluid filling a 3-dimensional horizontal domain bounded at the bottom by a smooth wall and at the top by a very rough wall. The latter consists in a plane wall covered with periodically distributed asperities which size...

By using the Hilbert uniqueness method (HUM), we study the exact controllability problem described by the wave equation in a three-dimensional horizontal domain bounded at the bottom by a smooth wall and at the top by a rough wall. The latter is assumed to consist in a plane wall covered with periodically distributed asperities whose size depends o...

We study a Dirichlet optimal control problem for a nonlinear monotone equation with degenerate weight function and with the coefficients which we adopt as controls in L ∞ (Ω). Since these types of equations can exhibit the Lavrentieff phenomenon, we consider the optimal control problem in coefficients in the so-called class of H-admissible solution...

We consider a mixed boundary-value problem for the Poisson equation in a plane two-level junction Ωε, which is the union of a domain Ω0 and a large number 2N of thin rods with variable thickness of order $\varepsilon =\mathcal{O}(N^{-1})$. The thin rods are divided into two levels depending on their length. In addition, the thin rods from each leve...

In this paper we deal with the homogenization problem for the Poisson equation in a singularly perturbed domain with multilevel periodically oscillating boundary. This domain consists of the body, a large number of thin cylinders joining to the body through the thin transmission zone with rapidly oscillating boundary. Inhomogeneous Fourier boundary...

In this paper we study a classical Dirichlet optimal control problem for a nonlinear elliptic equation with the coefficients as controls in $L^\infty(\Omega)$. Since such problems have no solutions in general, we make an assumption on the coefficients of the state equation and introduce the class of so-called solenoidal controls. Using the direct m...

In the paper, we deal with the homogenization problem for the Poisson equation in a singularly perturbed three-dimensional junction of a new type. This junction consists of a body and a large number of thin curvilinear cylinders, joining to body through a random transmission zone with rapidly oscillating boundary, periodic in one direction. Inhomog...

We study the asymptotic behavior of the solution of the Laplace equation in a domain perforated along the boundary. Assuming that the boundary microstructure is random, we construct the limit problem and prove the homogenization theorem. Moreover we apply those results to some spectral problems.

In the article we deal with the homogenization of a boundary-value problem for the Poisson equation in a singularly perturbed two-dimensional junction of a new type. This junction consists of a body and a large number of thin rods, which join the body through the random transmission zone with rapidly oscillating boundary. Inhomogeneous Fourier boun...

In this paper we study an optimal boundary control problem for the 3D steady-state Navier-Stokes equation in a cylindrically perforated domain. The control is the boundary velocity field supported on the 'vertical' sides of thin cylinders. We minimize the vorticity of viscous flow through thick perforated domain. We show that an optimal solution to...

We study the asymptotic behaviour of a parabolic optimal control problem in a domain Ω ε ⊂ℝ n , whose boundary ∂Ω ε contains a highly oscillating part. We consider this problem with two different classes of Dirichlet boundary controls, and, as a result, we provide its asymptotic analysis with respect to the different topologies of homogenization. I...

In this paper we study the asymptotic behaviour, as $\e$ tends to zero, of a class of boundary optimal control problems ${\mathbb{P}}_\e$, set in $\e$-periodically perforated domain. The holes have a critical size with respect to $\e$-sized mesh of periodicity. The support of controls is contained in the set of boundaries of the holes. This set is...

In this paper we study the asymptotic behaviour of a parabolic optimal control problem in a domain $\Omega_\e\subset\mathbb{R}^n$, whose boundary $\partial \Omega_\e$ contains a highly oscillating part. We consider this problem with two different classes of Dirichlet boundary controls, and, as a result, we provide its asymptotic analysis with respe...

We study the asymptotic behaviour of an optimal control problem for the Ukawa equation in a thick multi-structure with different types and classes of admissible boundary controls. This thick multi-structure consists of a domain (the junction's body) and a large number of epsilon-periodically situated thin cylinders. We consider two types of boundar...

We consider a perturbed initial/boundary-value problem for the heat equation in a thick multi-structure \Omega_\epsilon which is the union of a domain \Omega_0 and a large number N of \epsilon-periodically situated thin rings with variable thickness of order \epsilon = O(N^{-1}). The following boundary condition \partial_\nu u_\epsilon + \epsilon^\...

We consider a viscous incompressible flow in an infinite horizontal domain bounded at the bottom by a smooth wall and at the top by a rough wall. The latter is assumed to consist in a plane wall covered with periodically distributed asperities which size depends on a small parameter, and with a fixed height. We assume that the flow is governed by t...

We propose two different approaches for asymptotic analysis of the Neumann boundary-value problem for the Ukawa equation in a thick multistructure Ωε, which is the union of a domain Ω0 and a large number N of ε—periodically situated thin annular disks with variable thickness of order e = O( N - 1 )\varepsilon = \mathcal{O}\left( {N^{ - 1} } \right...

We consider a mixed boundary value problem for the Poisson equation in a thick multistructure Q,, which is the union of a domain Omega(0) and a large number N of epsilon-periodically situated thin rings with variable thickness of order epsilon = O(N-1). The Robin conditions are given on the lateral boundaries of the thin rings. The leading terms of...

In the present article we announce a result of the homogenization of a parabolic optimal control problem ℙ ε in thick multi-structures Ω ε ⊂ℝ n . Here by the thick multi-structure Ω ε we mean a domain Ω + and a large number of thin cylinders with axes parallel to Ox n and ε-periodically distributed along some manifold ∑ on the boundary of Ω + . We...

Using some special extension operator, a convergence theorem is proved for the solution to the Neumann boundary value problem for the Ukawa equation in a junction Ωε, which is the union of a domain Ω0 and a large number N of ε-periodically situated thin annular disks with variable thickness of order ε=(N-1), as ε → 0. Copyright © 2004 John Wiley &...

In this paper we study a mixed boundary value problem for the Poisson equation in a multi-structure Ω_ε, which is the union of a domain Ω_0 and a large number N of ε-periodically situated thin rings with variable thickness of order ε = O(N^{−1}). By using some special extension operator, we prove a convergence theorem as ε → 0 and investigate the a...

The leading terms of the asymptotic expansion for the solution to a mixed boundary value problem for the Poisson equation in a thick multi-structure, which is the union of some domain and a large number N of ε-periodically situated thin annular disks with variable thickness of order ε = O(N −1), are constructed and the corresponding estimates in th...

In this article we study the homogenization of an optimal control problem for a parabolic equation in a domain with highly oscillating boundary. We identify the limit problem, which is an optimal control problem for the homogenized equation and with a different cost functional.

We consider a mixed boundary-value problem for the Poisson equation in a plane two-level junction Ωε that is the union of a domain Ω0 and a large number 2N of thin rods with variable thickness of order ε = \({\cal O}\)(N −1). The thin rods are divided into two levels, depending on their length. In addition, the thin rods from each level are ε-perio...

We study the asymptotic behavior of the solution of the Laplace equation in a domain, a part of whose boundary is highly oscillating. The motivation comes from the study of a longitudinal flow in an infinite horizontal domain bounded at the bottom by a wall and at the top by a rugose wall. The latter is a plane covered with periodic asperities whos...

We study the asymptotic behavior of the solution of the Laplace equation in a domain, a part of whose boundary is highly oscillating. The motivation comes from the study of a longitudinal flow in an infinite horizontal domain bounded at the bottom by a wall and at the top by a rugose wall. The latter is a plane covered with periodic asperities whos...

In this paper we study the asymptotic behaviour of the Laplace equation in a periodically perforated domain of R n , where we assume that the period is ε and the size of the holes is of the same order of greatness. An homogeneous Dirichlet condition is given on the whole exterior boundary of the domain and on a flat portion of diameter \( \varepsi...

The functional F(u) = ∫B f(x, Du)dx is considered, where B is the unit ball in Rⁿ, u varies in the set of the locally Lipschitz functions on Rⁿ, and f belongs to a family of integrands containing, as model case, the following one f : (x, z) ∈ Rⁿ × Rⁿ → | < z, x > |/|x|ⁿ+|z|p, 1 < < n. The computation of the relaxed functional of F is provided. The...

We consider a Borel function g on ℝ n taking its values in [0,+∞], verify some weak hypothesis of continuity, such that ( dom g) ∘ =∅ and dom g is convex, and obtain an integral representation result for the lower semicontinuous envelope in the L 1 (ω)-topology of the integral functional G 0 (u 0 ,ω,u)=∫ ω g(∇u)dx, where u∈W (loc) (1,∞) (ℝ n ),u=u...

In the present paper we study the homogenization of variational problems for integral functionals with constraints on the gradient that describe some phenomena in elastic-plastic torsion theory.