Singular vanishing-viscosity limits of gradient flows: the finite-dimensional case

In this note we study the singular vanishing-viscosity limit of a gradient flow set in a finite-dimensional Hilbert space and driven by a smooth, but possibly non convex, time-dependent energy functional. We resort to ideas and techniques from the variational approach to gradient flows and rate-independent evolution to show that, under suitable assumptions, the solutions to the singularly perturbed problem converge to a curve of stationary points of the energy, whose behavior at jump points is characterized in terms of the notion of Dissipative Viscosity solution. We also provide sufficient conditions under which Dissipative Viscosity solutions enjoy better properties, which turn them into Balanced Viscosity solutions. Finally, we discuss the generic character of our assumptions.Comment: 27 page

Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature

In this paper we consider complete noncompact Riemannian manifolds $(M, g)$ with nonnegative Ricci curvature and Euclidean volume growth, of dimension $n \geq 3$. We prove a sharp Willmore-type inequality for closed hypersurfaces $\partial \Omega$ in $M$, with equality holding true if and only if $(M{\setminus}\Omega, g)$ is isometric to a truncated cone over $\partial\Omega$. An optimal version of Huisken's Isoperimetric Inequality for $3$-manifolds is obtained using this result. Finally, exploiting a natural extension of our techniques to the case of parabolic manifolds, we also deduce an enhanced version of Kasue's non existence result for closed minimal hypersurfaces in manifolds with nonnegative Ricci curvature.Comment: Any comment is welcome

Dimension reduction via Gamma convergence for soft active materials

We present a rigorous derivation of dimensionally reduced theories for thin sheets of nematic elastomers, in the finite bending regime. Focusing on the case of twist nematic texture, we obtain 2D and 1D models for wide and narrow ribbons exhibiting spontaneous flexure and torsion. We also discuss some variants to the case of twist nematic texture, which lead to 2D models with different target curvature tensors. In particular, we analyse cases where the nematic texture leads to zero or positive Gaussian target curvature, and the case of bilayers. \ua9 2017 Springer Science+Business Media Dordrech

Media devices in pre-school children: the recommendations of the Italian pediatric society

BACKGROUND: Young children are too often exposed to mobile devices (MD) and most of them had their own device. The adverse effects of a early and prolonged exposure to digital technology on pre-school children has been described by several studies. Aim of the study is to analyze the consequences of MD exposure in pre-school children. METHODS: We analyzed the documented effects of media exposure on children's mental and physical health. RESULTS: According to recent studies, MD may interfere with learning, children development, well being, sleep, sight, listening, caregiver-child relationship. DISCUSSION: Pediatricians should be aware of both the beneficial and side effects of MD and give advice to the families, according to children's age. CONCLUSION: In according to literature, the Italian Pediatric Society suggest that the media device exposure in childhood should be modulated by supervisors

COVID 19 vaccine in the pediatric age: the recommendation of the Italian Pediatric Society

Vaccine is an important and effective tool to protect from preventable infectious diseases. Neverthless, in the COVID-19 pandemic era, scientific and accurate information are required to responde to false and misleading information on efficacy and safety of immunization in the pediatric age