Optimal constants for a nonlocal approximation of sobolev norms and total variation

We consider the family of nonlocal and nonconvex functionals proposed and investigated by J. Bourgain, H. Brezis and H.-M. Nguyen in a series of papers of the last decade. It was known that this family of functionals Gamma-converges to a suitable multiple of the Sobolev norm or the total variation, depending on the summability exponent, but the exact constants and the structure of recovery families were still unknown, even in dimension 1. We prove a Gamma-convergence result with explicit values of the constants in any space dimension. We also show the existence of recovery families consisting of smooth functions with compact support. The key point is reducing the problem first to dimension 1, and then to a finite combinatorial rearrangement inequality

AIRBench: A DEA-based model for the benchmarking of airports revenues

The socio-economic development of countries is a key factor for the growth of people mobility, bringing an increase of inter and extra continental air passengers flows. On the other side, the leading model business of low cost companies is decreasing the revenues of airports coming from the avio operations. The two effects are increasing the awareness of airports management, historically focused on the avio operations, towards the mix of avio and commercial revenues. AIRBench is a DEA-based benchmark model developed by the ORO Group of Poltecnico di Torino and BDS, a consulting company specialized in the sector of airports management. Differently from other works in the literature, AIRBench uses data that can be obtained by publicly available documents and databases (consolidated balance sheets, web sites, and ENAC data) in order to link both avio and commercial revenues to the airports performances. The DEA-based method is applied to a panel of 21 airports (both Italian and European airports) in order to build a comparison of airports performances and to obtain a panel of reference management practices to evaluate additional airports. Moreover, given the airports considered efficient by the DEA models, a detailed analysis of causes, opportunities and threads for non-efficient airports is presented

Refined upper bounds on the coarsening rate of discrete, ill-posed diffusion equations

"We study coarsening phenomena observed in discrete, ill-posed diffusion equations that arise in a variety of applications, including computer vision, population dynamics and granular flow. Our results provide rigorous upper bounds on the coarsening rate in any dimension. Heuristic arguments and the numerical experiments we perform indicate that the bounds are in agreement with the actual rate of coarsening."http://deepblue.lib.umich.edu/bitstream/2027.42/64211/1/non8_12_002.pd

Finite difference approximation of the Mumford-Shah functional

We study the pointwise convergence and the Gamma-convergence of a family of nonlocal functionals defined in L^1_loc(R^n) to a local functional F(u) that depends on the gradient of u and on the set of discontinuity points of u. We apply this result to approximate a minimum problem introduced by Mumford and Shah to study edge detection in computer vision theory

Quasilinear degenerate parabolic equations of Kirchhoff type

We investigate the evolution problem for a degenerate parabolic equation of Kirchhoff type

Minimizing movements and evolution problems in Euclidean spaces

We study evolution curves of variational type, called minimizing movements, obtained via a time discretization and minimization method. We analyze examples in Euclidean spaces, where some classes of minimizing movements are solutions of suitable ordinary differential equations of gradient flow type. Finally, we construct an example to show that in general these evolution curves are not maximal slope curves

Singular perturbation hyperbolic-parabolic for degenerate nonlinear equations of Kirchhoff type

The theorem involving a locally Lipschitz continuous function is proven with a global-in-time uniform convergence result for an abstract setting of the initial-boundary value problem. The boundary value problem is a setting for the hyperbolic PDE with a nonlocal nonlinearity of Kirchhoff type. This equation is a model for the damped small transversal vibrations of an elastic string with fixed endpoints, and uniform density

Entire solutions of the one-dimensional Perona-Malik equation

We prove that every function u: R^{2} -> R of class C^{1}, satisfying the Perona-Malik equation for every (x,t) in R^{2}, is a stationary affine solution of the form u(x,t)=ax+b, where a and b are suitable real constants

Topological properties of attractors for dynamical systems

For a dynamical system {S_t} on a metric space X, we examine the question whether the topological properties of X are inherited by the global attractor A (if it exists). When {S_t} is jointly continuous, we prove that the Čech-Alexander-Spanier cohomology groups of A are isomorphic to the corresponding cohomology groups of X. The same conclusion is obtained in the case where {S_t} is a group and A has a bounded neighborhood which is a deformation retract of X