A singular perturbation problem in exact controllability of the Maxwell system

Abstract. This paper studies the exact controllability of the Maxwell system in a bounded domain, controlled by a current flowing tangentially in the boundary of the region, as well as the exact con-trollability the same problem but perturbed by a dissipative term multiplied by a small parameter in the boundary condition. This boundary perturbation improves the regularity of the problem and is therefore a singular perturbation of the original problem. The purpose of the paper is to examine the connection, for small values of the perturbation parameter, between observability estimates for the two systems, and between the optimality systems corresponding to the problem of norm minimum exact control of the solutions of the two systems from the rest state to a specied terminal state

Dynamics of thermoelastic thin plates: A comparison of four theories

Four distinct theories describing the flexural motion of thermoelastic thin plates are compared. The theories are due to Chadwick, Lagnese and Lions, Simmonds, and Norris. Chadwick's theory requires a 3D spatial equation for the temperature but is considered the most accurate as the others are derivable from it by different approximations. Attention is given to the damping of flexural waves. Analytical and quantitative comparisons indicate that the Lagnese and Lions model with a 2D temperature equation captures the essential features of the thermoelastic damping, but contains systematic inaccuracies. These are attributable to the approximation for the first moment of the temperature used in deriving the Lagnese and Lions equation. Simmonds' model with an explicit formula for temperature in terms of plate deflection is the simplest of all but is accurate only at low frequency, where the damping is linearly proportional to the frequency. It is shown that the Norris model, which is almost as simple as Simmond's, is as accurate as the more precise but involved theory of Chadwick.Comment: 2 figures, 1 tabl

Strichartz Estimates for the Vibrating Plate Equation

We study the dispersive properties of the linear vibrating plate (LVP) equation. Splitting it into two Schr\"odinger-type equations we show its close relation with the Schr\"odinger equation. Then, the homogeneous Sobolev spaces appear to be the natural setting to show Strichartz-type estimates for the LVP equation. By showing a Kato-Ponce inequality for homogeneous Sobolev spaces we prove the well-posedness of the Cauchy problem for the LVP equation with time-dependent potentials. Finally, we exhibit the sharpness of our results. This is achieved by finding a suitable solution for the stationary homogeneous vibrating plate equation.Comment: 18 pages, 4 figures, some misprints correcte

Thermodynamics of non-local materials: extra fluxes and internal powers

The most usual formulation of the Laws of Thermodynamics turns out to be suitable for local or simple materials, while for non-local systems there are two different ways: either modify this usual formulation by introducing suitable extra fluxes or express the Laws of Thermodynamics in terms of internal powers directly, as we propose in this paper. The first choice is subject to the criticism that the vector fluxes must be introduced a posteriori in order to obtain the compatibility with the Laws of Thermodynamics. On the contrary, the formulation in terms of internal powers is more general, because it is a priori defined on the basis of the constitutive equations. Besides it allows to highlight, without ambiguity, the contribution of the internal powers in the variation of the thermodynamic potentials. Finally, in this paper, we consider some examples of non-local materials and derive the proper expressions of their internal powers from the power balance laws.Comment: 16 pages, in press on Continuum Mechanics and Thermodynamic

Decline of gastric cancer mortality in common variable immunodeficiency in the years 2018-2022

Introduction: In patients with Common Variable Immunodeficiency, malignancy has been reported as the leading cause of death in adults, with a high risk of B-cell lymphomas and gastric cancer.Methods: We conducted a five-year prospective study aiming to update the incidence and mortality of gastric cancer and the incidence of gastric precancerous lesions in 512 CVID patients who underwent a total of 400 upper gastrointestinal endoscopies.Results: In the pre-pandemic period, 0.58 endoscopies were performed per patient/year and in the COVID-19 period, 0.39 endoscopies were performed per patient/year. Histology revealed areas with precancerous lesions in about a third of patients. Patients who had more than one gastroscopy during the study period were more likely to have precancerous lesions. Two patients received a diagnosis of gastric cancer in the absence of Helicobacter pylori infection. The overall prevalence of Helicobacter pylori infection in biopsy specimens was 19.8% and related only to active gastritis. Among patients who had repeated gastroscopies, about 20% progressed to precancerous lesions, mostly independent of Helicobacter pylori.Discussion: While gastric cancer accounted for one in five deaths from CVID in our previous survey, no gastric cancer deaths were recorded in the past five years, likely consistent with the decline in stomach cancer mortality observed in the general population. However, during the COVID-19 pandemic, cancer screening has been delayed. Whether such a delay or true decline could be the reason for the lack of gastric cancer detection seen in CVID may become clear in the coming years. Due to the high incidence of precancerous lesions, we cannot rely on observed and predicted trends in gastric cancer mortality and strongly recommend tailored surveillance programs

Boundary stabilization of numerical approximations of the 1-D variable coefficients wave equation: A numerical viscosity approach

In this paper, we consider the boundary stabilization problem associated to the 1- d wave equation with both variable density and diffusion coefficients and to its finite difference semi-discretizations. It is well-known that, for the finite difference semi-discretization of the constant coefficients wave equation on uniform meshes (Tébou and Zuazua, Adv. Comput. Math. 26:337–365, 2007) or on somenon-uniform meshes (Marica and Zuazua, BCAM, 2013, preprint), the discrete decay rate fails to be uniform with respect to the mesh-size parameter. We prove that, under suitable regularity assumptions on the coefficients and after adding an appropriate artificial viscosity to the numerical scheme, the decay rate is uniform as the mesh-size tends to zero. This extends previous results in Tébou and Zuazua (Adv. Comput.Math. 26:337–365, 2007) on the constant coefficient wave equation. The methodology of proof consists in applying the classical multiplier technique at the discrete level, with a multiplier adapted to the variable coefficients

Safety of mRNA COVID-19 Vaccines in Patients with Inborn Errors of Immunity: an Italian Multicentric Study

Purpose: Little is known about vaccine safety in inborn errors of immunity (IEI) patients during the current vaccination campaign for COVID-19. To better investigate the reactogenicity and adverse event profile after two, three, and four doses of mRNA vaccines, we conducted an observational, multicentric study on 342 PID patients from four Italian Referral Centres. Methods: We conducted a survey on self-reported adverse reactions in IEI patients who received mRNA vaccine by administering a questionnaire after each dose. Results: Over the whole study period, none of the patients needed hospitalization or had hypersensitivity reactions, including anaphylaxis and delayed injection site reaction. After two vaccination doses, 35.4% of patients showed only local reactogenicity-related symptoms (RrS), 44.4% reported both systemic and local RrS, and 5% reported only systemic RrS. In more than 60% of cases, local or systemic RrS were mild. After the first and second booster doses, patients showed fewer adverse events (AEs) than after the first vaccination course. Patients aged 50 years and older reported adverse events and RrS less frequently. Among AEs requiring treatment, one common variable immune deficiency patient affected by T cell large granular lymphocytic leukemia developed neutropenia and one patient had Bell’s paralysis perhaps during herpes zoster reactivation. Conclusion: Although our follow-up period is relatively short, the safety data we reported are reassuring. This data would help to contrast the vaccine hesitancy often manifested by patients with IEI and to better inform their healthcare providers

Control and stabilization of waves on 1-d networks

We present some recent results on control and stabilization of waves on 1-d networks.The fine time-evolution of solutions of wave equations on networks and, consequently, their control theoretical properties, depend in a subtle manner on the topology of the network under consideration and also on the number theoretical properties of the lengths of the strings entering in it. Therefore, the overall picture is quite complex.In this paper we summarize some of the existing results on the problem of controllability that, by classical duality arguments in control theory, can be reduced to that of observability of the adjoint uncontrolled system. The problem of observability refers to that of recovering the total energy of solutions by means of measurements made on some internal or external nodes of the network. They lead, by duality, to controllability results guaranteeing that L 2-controls located on those nodes may drive sufficiently smooth solutions to equilibrium at a final time. Most of our results in this context, obtained in collaboration with R. Dáger, refer to the problem of controlling the network from one single external node. It is, to some extent, the most complex situation since, obviously, increasing the number of controllers enhances the controllability properties of the system. Our methods of proof combine sidewise energy estimates (that in the particular case under consideration can be derived by simply applying the classical d'Alembert's formula), Fourier series representations, non-harmonic Fourier analysis, and number theoretical tools.These control results belong to the class of the so-called open-loop control systems.We then discuss the problem of closed-loop control or stabilization by feedback. We present a recent result, obtained in collaboration with J. Valein, showing that the observability results previously derived, regardless of the method of proof employed, can also be recast a posteriori in the context of stabilization, so to derive explicit decay rates (as) for the energy of smooth solutions. The decay rate depends in a very sensitive manner on the topology of the network and the number theoretical properties of the lengths of the strings entering in it.In the end of the article we also present some challenging open problems