Gianni Gilardi

Magnetostatic and Electrostatic Problems in Inhomogeneous Anisotropic Media with Irregular Boundary and Mixed Boundary Conditions

Magnetostatic and electrostatic problems with mixed boundary conditions are studied. The medium can have a nonsmooth boundary and very irregular physical properties due to inhomogeneity and anisotropy. The topological assumptions are general enough to meet the requirements of the engineering applications. Necessary and sufficient conditions for solvability are found and the set of the solutions is characterized. Moreover, uniqueness is recovered by means of a finite number of supplementary conditions which are equivalent to prescribing a finite number of suitably chosen fluxes or potentials. A functional framework in which other important problems of electromagnetics naturally fit is developed.

On a model for phase separation in binary alloys driven by mechanical effects

This work is concerned with the mathematical analysis of a system of partial differential equations modeling the effect of phase separation driven by mechanical actions in binary alloys like tin/lead solders. The system combines the (quasistationary) balance of linear momentum with a fourth order evolution equation of Cahn\_Hilliard type for the phase separation, and it is highly nonlinearly coupled. Existence and uniqueness results are shown. TEL:: 0382505631 EMAIL:: bonetti@dimat.unipv.it

Well-posedness and Long-time Behavior for a Nonstandard Viscous Cahn-Hilliard System

We study a diffusion model of phase field type, consisting of a system of two partial differential equations encoding the balances of microforces and microenergy; the two unknowns are the order parameter and the chemical potential. By a careful development of uniform estimates and the deduction of certain useful boundedness properties, we prove existence and uniqueness of a global-in-time smooth solution to the associated initial/boundary-value problem; moreover, we give a description of the relative

On the Cahn–Hilliard equation with dynamic boundary conditions and a dominating boundary potential

\omega

A boundary control problem for the pure Cahn–Hilliard equation with dynamic boundary conditions

-limit set.

EXISTENCE AND UNIQUENESS OF A GLOBAL-IN-TIME SOLUTION TO A PHASE SEGREGATION PROBLEM OF THE ALLEN–CAHN TYPE

The Cahn–Hilliard and viscous Cahn–Hilliard equations with singular and possibly nonsmooth potentials and dynamic boundary condition are considered and some well-posedness and regularity results are proved.

Analysis of a time discretization scheme for a nonstandard viscous Cahn-Hilliard system

Abstract A boundary control problem for the pure Cahn–Hilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first-order necessary conditions for optimality are proved.

Global Existence for a Nonstandard Viscous Cahn--Hilliard System with Dynamic Boundary Condition

We study a model of phase segregation of the Allen–Cahn type, consisting in a system of two differential equations, one partial and the other ordinary, respectively interpreted as balances of microforces and microenergy; the two unknowns are the order parameter entering the standard A–C equation and the chemical potential. We introduce a notion of maximal solution to the o.d.e., parametrized on the order-parameter field; and, by substitution in the p.d.e. of the so-obtained chemical potential field, we give the latter equation the form of an Allen–Cahn equation for the order parameter, with a memory term. Finally, we prove the existence and uniqueness of global-in-time smooth solutions to this modified A–C equation, and we give a description of the relative ω-limit set.

Sliding Mode Control for a Nonlinear Phase-Field System

In this paper we propose a time discretization of a system of two parabolic equations describing diffusion-driven atom rearrangement in crystalline matter. The equations express the balances of microforces and microenergy; the two phase fields are the order parameter and the chemical potential. The initial and boundary-value problem for the evolutionary system is known to be well posed. Convergence of the discrete scheme to the solution of the continuous problem is proved by a careful development of uniform estimates, by weak compactness and a suitable treatment of nonlinearities. Moreover, for the difference of discrete and continuous solutions we prove an error estimate of order one with respect to the time step.

Weak solution to hyperbolic Stefan problems with memory

In this paper, we study a model for phase segregation taking place in a spatial domain that was introduced by Podio-Guidugli [Ric. Mat., 55 (2006), pp. 105--118]. The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur, that are difficult to handle analytically. In contrast to the existing literature about this PDE system, we consider here a dynamic boundary condition involving the Laplace--Beltrami operator for the order parameter. This boundary condition models an additional nonconserving phase transition occurring on the surface of the domain. Different well-posedness results are shown, depending on the smoothness properties of the involved bulk and surface free energies.