Ricerc@Sapienza

In \cite{LM} we considered a nontrivial example of eigenfunction in the sense of distribution for the planar Fourier transform. Here a method to obtain other eigenfunctions is proposed.
Moreover we consider positive homogeneous distributions in \(\R^n\) of order \(-n/2\). It is shown that \({F(\om)}{|\bx|^{-n/2}}, |\om|=1\) is an eigenfunction in the sense of distribution of the Fourier transform if and only if \(F(\om)\) is an eigenfunction of a certain singular integral operator on the unit sphere of \(\R^n\).

In this paper we will consider oscillations of square viscoelastic membranes by adding to the wave equation another term, which takes into account the memory. To this end, we will study a class of integrodifferential equations in square domains. By using accurate estimates of the spectral properties of the integrodifferential operator, we will prove an inverse observability inequality.

We solve the reachability problem for a coupled wave-wave system with an integro-differential term. The control functions act on one side of the boundary. The estimates on the time is given in terms of the parameters of the problem and they are explicitly computed thanks to Ingham type results. Nevertheless some restrictions appear in our main results. The Hilbert Uniqueness Method is briefly recalled. Our findings can be applied to concrete examples in viscoelasticity theory

The present article discusses the exact observability of the wave equation when the obser- vation subset of the boundary is variable in time. In the one-dimensional case, we prove an equivalent condition for the exact observability, which takes into account only the location in time of the obser- vation. To this end we use Fourier series. Then we investigate the two specific cases of single exchange of the control position, and of exchange at a constant rate. In the multi-dimensional case, we analyse sufficient conditions for the exact observability relying on the multiplier method.

We investigate the simultaneous observability of infinite systems of vibrating strings or beams having a common endpoint where the observation is taking place. Our results are new even for finite systems because we allow the vibrations to take place in independent directions. Our main tool is a vectorial generalization of some classical theorems of Ingham, Beurling and Kahane in nonharmonic analysis.

We study the observability of the wave equation when the
observation set changes over time.
For the one dimensional Neumann problem, using Fourier series, we
are able to prove for the exact observability an equivalent condition
already known for the Dirichlet problem; see [1].
For the observability problem with Dirichlet boundary conditions, we
focus on multidimensional problems and the observability inequality is
proven through a multiplier approach.
Besides this, we present some applications and a numerical simulation.

We investigate the internal observability of the wave equation with Dirichlet boundary conditions in tilings. The paper includes a general result relating internal observability problems in general domains to their tiles, and a discussion of the case in which the domain is the 30-60-90 triangle.