Evolution equations on non flat waveguides

We investigate the dispersive properties of evolution equations on waveguides with a non flat shape. More precisely we consider an operator $H=-\Delta_{x}-\Delta_{y}+V(x,y)$ with Dirichled boundary condition on an unbounded domain $\Omega$, and we introduce the notion of a \emph{repulsive waveguide} along the direction of the first group of variables $x$. If $\Omega$ is a repulsive waveguide, we prove a sharp estimate for the Helmholtz equation $Hu-\lambda u=f$. As consequences we prove smoothing estimates for the Schr\"odinger and wave equations associated to $H$, and Strichartz estimates for the Schr\"odinger equation. Additionally, we deduce that the operator $H$ does not admit eigenvalues.Comment: 22 pages, 4 figure

Spectral enclosures for the damped elastic wave equation

In this paper we investigate spectral properties of the damped elastic wave equation. Deducing a correspondence between the eigenvalue problem of this model and the one of Lamé operators with non self-adjoint perturbations, we provide quantitative bounds on the location of the point spectrum in terms of suitable norms of the damping coefficient

Uniform resolvent estimates and absence of eigenvalues for Lamé operators with complex potentials

We consider the $0$-order perturbed Lamé operator $-\Delta^\ast + V(x)$. It is well known that if one considers the free case, namely $V=0,$ the spectrum of $-\Delta^\ast$ is purely continuous and coincides with the non-negative semi-axis. The first purpose of the paper is to show that, at least in part, this spectral property is preserved in the perturbed setting. Precisely, developing a suitable multipliers technique, we will prove the absence of point spectrum for Lamé operator with potentials which satisfy a variational inequality with suitable small constant. We stress that our result also covers complex-valued perturbation terms. Moreover the techniques used to prove the absence of eigenvalues enable us to provide uniform resolvent estimates for the perturbed operator under the same assumptions about $V$

Uniform resolvent estimates and absence of eigenvalues of biharmonic operators with complex potentials

We quantify the subcriticality of the bilaplacian in dimensions greater than four by providing explicit repulsivity/smallness conditions on complex additive perturbations under which the spectrum remains stable. Our assumptions cover critical Rellich-type potentials too. As a byproduct we obtain uniform resolvent estimates in weighted spaces. Some of the results are new also in the self-adjoint setting

The Kirchhoff Equation with Global Solutions in Unbounded Domains

The aim of this paper is to find a general class of data in which the global well-posedness for the initial-boundary value problem to the Kirchhoff equation in unbounded domains is assured. The result obtained in the present paper will be applied to the existence of scattering operators. Some examples of function spaces contained in this class will be presented

Smoothing Strichartz Estimates for Dispersive Equations Perturbed by a First Order Differential Operator

L’intento di questa tesi e’ quello di presentare in maniera esauriente nuove tecniche sviluppate per risolvere alcuni problemi aperti nel campo delle equazioni alle derivate parziali di tipo iperbolico. Piu’ precisamente vengono dimostrate nuove stime a priori di tipo smoothing-Strichartz utilizzate poi per ottenere stabilita’ per classi di equazioni dispersive ed onde solitarie perturbate da potenziali di tipo magnetico con particolari condizioni di decadimento sui coefficienti del potenziale stesso(short range). Tale tecnica e’ utile anche per trattare perturbazioni di tipo operatore differenziale di ordine uno(come risulta essere un potenziale di tipo magnetico). Vengono altresì dimostrate nuove stime per la ”risolvente libera” e per quella ”perturbata”,utilizzando tecniche che vengono dall’analisi armonica e quella funzionale. Viene anche affrontato il problema degli autovalori per particolari tipi di operatori, con applicazione alla teoria delle risonanze. Nella tesi vengono presentati nuovi risultati pubblicati su riviste di matematica