Alfredo Lorenzi

An inverse problem for a semilinear parabolic equation

SummaryIn this paper we are concerned with the study of the stability of an unknown non-linear term in a parabolic equation in dependence on over specified Cauchy-Dirichlet data prescribed on the parabolic boundary of the open set under consideration. Since, in general, the dependence of the nonlinear term upon the data is not stable with respect to L∞ -metrics, we show how a Hölder continuity may be restored under mild restrictions for the set of admissible solutions.

A stability result via Carleman estimates for an inverse source problem related to a hyperbolic integro-differential equation

First we prove a Carleman estimate for a hyperbolic integro-differential equation. Next we apply such a result to identify a spatially dependent function in a source term by an (additional) single measurement on the boundary.

Identification problems for parabolic delay differential equations with measurement on the boundary

This paper deals with the problem of recovering a scalar time-dependent function in the source term in an abstract parabolic equation with delay. Both the so-called singular and nonsingular cases are considered. For such problems existence and uniqueness results as well as continuous dependence upon the data are proved. Applications to partial differential equations, with measurement of the flux on the boundary, are given.

An inverse hyperbolic integrodifferential problem arising in geophysics II

eG3, t) = g(t), t E K4 Tl, (1.4) x0 being a fixed point in IF’. The problem of determining h arises in Geophysics (see [ 1, 21) and was studied by Yanno [ 1 l] for the one-dimensional equation, using a supplementary information different from (1.4). In [5] we proved local (in time) existence, global uniqueness and stability for the inverse problem (1 . l)-(1.4) by means of a technique very different from that used in [ 111. Here we establish analogous results holding also for a general (bounded) geometry.

On elliptic equations with piecewise constant coefficients

In this work we prove an existence and uniqueness theorem for solutions in 2(Rt) of second order elliptic equations with coefficients that are constant on the half-spaces R+n and R n

An L^{p}-approach to singular linear parabolic equations in bounded domains

Singular means here that the parabolic equation is not in normal form neither can it be reduced to such a form. For this class of problems, following the operator approach used in [1], we prove global in time existence and uniqueness theorems related to (spatial) -spaces. Various improvements to [2], [3] are given.

Recovering a Lamé kernel in a viscoelastic system

In the equation of viscoelasticity related to a geometric cylinder, we recover the spatial part p of a factorized Lamé kernel depending on two spatial sectional variables. More explicitly, we determine sufficient conditions on the data ensuring the (local) uniqueness of p and its continuous dependence on the data in suitable metrics. §A. Lorenzi and F. Messina are members of the research group G.N.A.M.P.A. of the Italian Instiuto Nazionale di Alta Matematica (INdAM).

Identification problems related to electro-magneto — elastic interactions

We study the interactions of a vibrating electroconductive elastic body and the electromagnetic field arising under the influence of the elastic field. In this paper we neglect both the effect of the electromagnetic field on the process of the elastic wave propagation and any convection current. We assume as well that the motion of the continuum occurs at velocities that are much smaller than the propagation velocity of the electromagnetic waves through the elastic medium. We prove an existence and uniqueness result both for the direct problem and the inverse one. which consists in identifying the unknown scalar coefficient f ( t ) in the body density force f ( t ) g ( t y x ) acting on the elastic body, when some additional measurement is available.

Direct and inverse problems related to MEMS

This paper deals with direct and inverse evolution problems which come up in studying micro-electro-mechanical-systems: here we consider a nonlinear and nonlocal MEMS model. The inverse problem consists of recovering a time-varying Coulomb potential by exploiting some accessible measurements, which depend on the dynamic displacement of the system. Local existence, uniqueness and continuous dependence results are proved for both direct and inverse problems.

An inverse problem in potential theory

SummaryWe consider the classical problem of finding the density ϱ of a material body Ω embedded into a region S, when the potential generated by Ω (possibly coinciding with S) is known outside (or on the surface of) S. In the set of such solutions we look for the density