Ferruccio Colombini

Uniqueness of continuous solutions for BV vector fields

We consider a vector field whose coefficients are functions of bounded variation, with a bounded divergence. We prove the uniqueness of continuous solutions for the Cauchy problem.

Time-Dependent Loss of Derivatives for Hyperbolic Operators with Non Regular Coefficients

In this paper we will study the Cauchy problem for strictly hyperbolic operators with low regularity coefficients in any space dimension N ≥ 1. We will suppose the coefficients to be log-Zygmund continuous in time and log-Lipschitz continuous in space. Paradifferential calculus with parameters will be the main tool to get energy estimates in Sobolev spaces and these estimates will present a time-dependent loss of derivatives.

Well-posedness of the cauchy problem in gevrey classes for some weakly hyperbolic equations of higher order

This article is devoted to the study of the Cauchy problem in Gevrey classes for some higher order weakly hyperbolic equations with time-dependent coefficients and without lower order terms.

The Well-Posedness Issue in Sobolev Spaces for Hyperbolic Systems with Zygmund-Type Coefficients

In this paper we study the well-posedness of the Cauchy problem for first order hyperbolic systems with constant multiplicities and with low regularity coefficients depending just on the time variable. We consider Zygmund and log-Zygmund type assumptions, and we prove well-posedness in H∞ respectively without loss and with finite loss of derivatives. The key to obtain the results is the construction of a suitable symmetrizer for our system, which allows us to recover energy estimates (with or without loss) for the hyperbolic operator under consideration. This can be achievied, in contrast with the classical case of systems with smooth (say Lipschitz) coefficients, by adding one step in the diagonalization process, and building the symmetrizer up to the second order.

Weyl formula for the negative dissipative eigenvalues of Maxwell’s equations

Let

Well-posedness of the Cauchy problem for a hyperbolic equation with non-Lipschitz coefficients

V(t) = e^{tG_b},, t ge 0,

The global Cauchy problem for a vibrating beam equation

V(t)=etGb,t≥0, be the semigroup generated by Maxwell’s equations in an exterior domain