A pathological example in Nonlinear Spectral Theory

We construct an open set $\Omega\subset\mathbb{R}^N$ on which an eigenvalue problem for the $p-$Laplacian has not isolated first eigenvalue and the spectrum is not discrete. The same example shows that the usual Lusternik-Schnirelmann minimax construction does not exhaust the whole spectrum of this eigenvalue problem.Comment: 9 pages, 1 figur

An overdetermined problem in 2D linearised hydrostatics

In two spatial dimensions, we discuss the relation between the solvability of Schiffer's overdetermined problem and the optimality, among sets of prescribed area, of the first eigenvalue in the buckling problem for a clamped plate and that of the first eigenvalue of the Stokes operator. For the latter, we deduce that the minimisers under area constraint that are smooth and simply connected must be discs from the fact that a pressureless velocity is a necessary condition of optimality

On the Hong-Krahn-Szego inequality for the p−p-Laplace operator

18 pagesGiven an open set $\Omega$, we consider the problem of providing sharp lower bounds for $\lambda_2(\Omega)$, i.e. its second Dirichlet eigenvalue of the $p-$Laplace operator. After presenting the nonlinear analogue of the {\it Hong-Krahn-Szego inequality}, asserting that the disjoint unions of two equal balls minimize $\lambda_2$ among open sets of given measure, we improve this spectral inequality by means of a quantitative stability estimate. The extremal cases $p=1$ and $p=\infty$ are considered as well

An overview on constrained critical points of Dirichlet integrals

We consider a natural generalization of the eigenvalue problem for the Laplacian with homogeneous Dirichlet boundary conditions. This corresponds to look for the critical values of the Dirichlet integral, constrained to the unit $L^q$ sphere. We collect some results, present some counter-examples and compile a list of open problems.Comment: 41 pages, 5 figures. This paper evolved from a set of notes for a talk delivered by the first author at the workshop "Nonlinear Meeting in Turin 2019

Transmission conditions obtained by homogenisation

Given a bounded open set in Rn, n 652, and a sequence (Kj) of compact sets converging to an (n-1)-dimensional manifold M, we study the asymptotic behaviour of the solutions to some minimum problems for integral functionals on \u3a9\Kj, with Neumann boundary conditions on 02(\u3a9\Kj). We prove that the limit of these solutions is a minimiser of the same functional on \u3a9\M subjected to a transmission condition on M, which can be expressed through a measure \ub5 supported on M. The class of all measures that can be obtained in this way is characterised, and the link between the measure \ub5 and the sequence (Kj) is expressed by means of suitable local minimum problems

Normal approximation of Random Gaussian Neural Networks

In this paper we provide explicit upper bounds on some distances between the (law of the) output of a random Gaussian NN and (the law of) a random Gaussian vector. Our results concern both shallow random Gaussian neural networks with univariate output and fully connected and deep random Gaussian neural networks, with a rather general activation function. The upper bounds show how the widths of the layers, the activation functions and other architecture parameters affect the Gaussian approximation of the ouput. Our techniques, relying on Stein's method and integration by parts formulas for the Gaussian law, yield estimates on distances which are indeed integral probability metrics, and include the total variation and the convex distances. These latter metrics are defined by testing against indicator functions of suitable measurable sets, and so allow for accurate estimates of the probability that the output is localized in some region of the space. Such estimates have a significant interest both from a practitioner's and a theorist's perspective

What's in a Name? Shifting Identities of Traditional Organized Crime in Canada in the Transnational Fight against the Calabrian ‘Ndrangheta

The Italian antimafia authorities have warned Canadian law enforcement about the risks and the growing concerns for the infiltration of clans of the Calabrian mafia, known as ‘ndrangheta, in Eastern Canada. The alarm linked to the rise of the ‘ndrangheta challenges the paradigms of traditional organized crime in Canada, because the ‘ndrangheta is presented as traditional but also innovative and more pervasive than other mafia-type groups. Through access to confidential investigations and interviews to key specialist law enforcement teams in Toronto and Montreal, this article investigates today's institutional perception of mafia – the ‘ndrangheta in particular – in Canada when compared to Italian conceptualizations. I will argue that the changes in narratives in Canada can be read in relation to changes in the Italian identity in the country, moving towards regionalization and specialist knowledge of ethnic differences

Existence, Uniqueness, Optimization and Stability for low Eigenvalues of some Nonlinear Operators

The thesis surveys some recent results obtained in the field of nonlinear partial differential equations and calculus of variations about eigenvalues of nonlinear operators