Roberto Gianni

Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues

We study a problem set in a finely mixed periodic medium, modelling electrical conduction in biological tissues. The unknown electric potential solves standard elliptic equations set in different conductive regions (the intracellular and extracellular spaces), separated by a dielectric surface (the cell membranes), which exhibits both a capacitive and a nonlinear conductive behaviour. Accordingly, dynamical conditions prevail on the membranes, so that the dependence of the solution on the time variable t is not only of parametric character. As the spatial period of the medium goes to zero, the electric potential approaches in a suitable sense a homogenization limit u0, which keeps the prescribed boundary data, and solves the equation . This is an elliptic equation containing a term depending on the history of the gradient of u0; the matrices B0, A1 in it depend on the microstructure of the medium. More exactly, we have that, in the limit, the current is still divergence-free, but it depends on the history of the potential gradient, so that memory effects explicitly appear. The limiting equation also contains a term ℱ keeping trace of the initial data.

Classical solutions to a multidimensional free boundary problem arising in combustion theory

In this paper the authors consider a free boundary problem arising in the mathematical theory of combustion. Classical solutions are examined to fairly general class of such problems, posed in N space dimensions, N [ge] 1. A model problem (to be completed with suitable initial and boundary conditions) is u[sub t] [minus] [Delta]u = 0, in G[sub T]/S[sub T]; u = 0, on S[sub T]; [D[sub u]] = 1, on S[sub T] where G [contained in] R[sup N] is an open set, G[sub T] = G x (0,T). Here u has the meaning of a scaled temperature, and the free boundary S[sub T] is a smooth surface of R[sup N] x R[sup +], corresponding to the flame front and separating the burnt region from the unburnt one. The authors prove existence and uniqueness for small times of a classical solution to a class of free boundary problems. The results hold for both one and two-phase problems. 10 refs.

The semilinear heat equation with a Heaviside source term

We consider the initial value problem for the equation u t = u xx +H(u), where H is the Heaviside graph, on a bounded interval with Dirichlet boundary conditions, and discuss existence, regularity and uniqueness of solutions and interfaces .

Homogenization limit for electrical conduction in biological tissues in the radio-frequency range

We study an evolutive model for electrical conduction in biological tissues, where the conductive intra-cellular and extracellular spaces are separated by insulating cell membranes. The mathematical scheme is an elliptic problem, with dynamical boundary conditions on the cell membranes. The problem is set in a nely mixed periodic medium. We show that the homogenization limit u0 of the electric potential, obtained as the period of the microscopic structure approaches zero, solves the equation

Continuous dependence on the constitutive functions for a class of problems describing fluid flow in porous media

In this paper we consider the PDE describing the fluid flow in a porous medium, focusing on the solution’s dependence upon the choice of the saturation curve and the hydraulic conductivity. Basically, we consider two different saturation curves (say θ1 and θ2) and two different hydraulic conductivities (K1 and K2) which are both “close” in the Lloc-norm. Then we find estimates to prove a constitutive stability for the solutions of the corresponding problems with the same boundary and initial conditions.

Exponential asymptotic stability for an elliptic equation with memory arising in electrical conduction in biological tissues

We study an electrical conduction problem in biological tissues in the radiofrequency range, which is governed by an elliptic equation with memory. We prove the time exponential asymptotic stability of the solution. We provide in this way both a theoretical justification to the complex elliptic problem currently used in electrical impedance tomography and additional information on the structure of the complex coefficients appearing in the elliptic equation. Our approach relies on the fact that the elliptic equation with memory is the homogenisation limit of a sequence of problems for which we prove suitable uniform estimates.

Global existence of a classical solution for a large class of free boundary problems in one space dimension

We prove “global” classical solvability and a uniqueness theorem for a wide class of one dimensional free boundary problems in which the evolution of the free boundarys(t) is controlled by the derivatives up to the second order ofu (u together withs(t) is an unknown of our problem) and by a functional ofs(t) itself. In the Introduction it is showed the physical background from which such problems arise.

A free boundary problem in an absorbing porous material with saturation dependent permeability

Abstract. We prove a well posedness result for a free boundary problem describing the filtration of an incompressible viscous fluid in a porous medium containing water absorbing granules.¶The location of the wetting front (the free boundary) is described by a Stefan like problem for a parabolic equation which is coupled with an hyperbolic equation describing the absorption kinetic of the granules.

The Constant Flow Rate Problem for Fluids with Increasing Yield Stress in a Pipe

Experiments show that the degradation effect observed during both stirring and pipelining tests of some coal-water slurries is mainly to be ascribed to the increase of the yield stress. Regardless of the particular mathematical model adopted to investigate the dynamics of these fluids, engineering applications force us to consider the problem of how long a constant flow rate can be maintained during the pipelining process. We choose a Bingham model where the yield stress is assumed to increase with the dissipated energy as in [5]. It is first shown that the constant flow rate problem is equivalent to solving a nonlinear functional equation in the unknown pressure gradient that generalizes the classical algebraic Buckingham equation for the same problem with constant rheological parameters. By means of a fixed-point argument we also prove that the functional equation has one and only one solution which is local in time. We finally find an estimate from below of the interval of the interval of existence. Numerical results are rather good and agree with those expected from the engineering point of view.

A REACTION INFILTRATION PROBLEM: EXISTENCE, UNIQUENESS, AND REGULARITY OF SOLUTIONS IN TWO SPACE DIMENSIONS

In this paper, we consider a reaction infiltration problem consisting of a parabolic equation for the concentration, an elliptic equation for the pressure, and an ordinary differential equation for the porosity. We establish global existence, uniqueness and regularity of the solution in a two-dimensional finite strip (−M, M)×(0, 1) and the existence and partial regularity of solutions in an infinite strip (−∞, ∞)×(0, 1).