In recent years a number of variational models related to reconstruction problems in computer vision have been proposed.

We study the convergence and decay rate to equilibrium of bounded solutions of the quasilinear parabolic equation ut - div a(x , ? u) + f (x , u) = 0 on a bounded domain, subject to Dirichlet boundary and to initial conditions. The data are supposed to satisfy suitable regularity and growth conditions. Our approach to the convergence result and decay estimate is based on the ?ojasiewicz-Simon gradient inequality which in the case of the semilinear heat equation is known to give optimal decay estimates. The abstract results and their applications are discussed also in the framework of Orlicz-Sobolev spaces.

We prove a semi-continuity theorem for an integral functional made up by a polyconvex energy and a surface term. Our result extends a well-known result by Ball to the BV framework.

We show the Gamma-convergence of a family of discrete functionals to the Mumford and Shah image segmentation functional. The functionals of the family are constructed by modifying the elliptic approximating functionals proposed by Ambrosio and Tortorelli. The quadratic term of the energy related to the edges of the segmentation is replaced by a nonconvex functional.

The objective of this paper is the derivation and the analysis of a simple explicit numerical scheme for general one-dimensional filtration equations. It is based on an alternative formulation of the problem using the pseudoinverse of the density's repartition function. In particular, the numerical approximations can be proven to satisfy a contraction property for a Wasserstein metric. Various numerical results illustrate the ability of this numerical process to capture the time-asymptotic decay towards self-similar solutions even for fast-diffusion equations.

Abstract. One-dimensional electronic conduction is investigated in a special case usually referred to as the harmonic crystal, meaning essentially that atoms are assumed to move like coupled harmonic oscillators within the Born–Oppenheimer approximation. We recall their dispersion relation and derive a WKB system approximately satisfied by any electron's wavefunction inside a given energy band. This is then numerically solved according to the method of K-branch solutions. Numerical results are presented in the case where atoms move with one- or two-modes vibrations; finally, we include the case where the Poisson self-interaction potential also influences the electrons' dynamics.

An extended FitzHugh-Nagumo model coupled with dynamical heat transfer in tissue, as described by a bioheat equation, is derived and confronted with experiments. The main outcome of this analysis is that traveling pulses and spiral waves of electric activity produce temperature variations on the order of tens of 0°C. In particular, the model predicts that a spiral wave’s tip, heating the surrounding medium as a consequence of the Joule effect, leads to characteristic hot spots. This process could possibly be used to have a direct visualization of the tip’s position by using thermal detectors.

The definition of relative accelerations and strains among a set of comoving particles is studied in connection with the geometric properties of the frame adapted to a \lq\lq fiducial observer." We find that a relativistically complete and correct definition of strains must take into account the transport law of the chosen spatial triad along the observer's congruence. We use special congruences of (accelerated) test particles in some familiar spacetimes to elucidate such a point. The celebrated idea of Szekeres' compass of inertia, arising when studying geodesic deviation among a set of free-falling particles, is here generalized to the case of accelerated particles. In doing so we have naturally contributed to the theory of relativistic gravity gradiometer. Moreover, our analysis was made in an observer-dependent form, a fact that would be very useful when thinking about general relativistic tests on space stations orbiting compact objects like black holes and also in other interesting gravitational situations.

The accurate calibration of thousands of thermometers involves the optimzation of the experimental design to save time and costs. A modelling procedure is proposed for the construction of a suitable free-knots cubic spline model. An iterative algorithm is used and a minimum experimental design is identified.

The pulsatile flow in a curved elastic pipe of circular cross section is investigated. The unsteady flow of a viscous fluid and the wall motion equations are written in a toroidal coordinate system, superimposed and linearized over a steady state solution. Being the main application relative to the vascular system, the radius of the pipe is assumed small compared withthe radius of curvature. This allows an asymptotic analysis over thecurvature parameter. The model results an extension of the Womersley's model for the straight elastic tube. A numerical solution is found for the first order approximation and computational results are finally presented, demonstrating the role of curvature in the wave propagation and in the development of a secondary flow.

We consider a thin metallic plate whose top side is inaccessible and in contact with a corroding fluid. Heat exchange between metal and fluid follows linear Newton's cooling law as long as the inaccessible side is not damaged. We assume that the effects of corrosion are modeled by means of a nonlinear perturbation in the exchange law. On the other hand, we are able to heat the conductor and take temperature maps of the accessible side. Our goal is to recover the nonlinear perturbation of the exchange law on the top side from thermal data collected on the opposite one (thermal imaging). In this paper we use a stationary model, i.e., the temperature inside the plate is assumed to fulfill Laplace's equation. Hence, our problem is stated as an inverse ill-posed problem for Laplace's equation with nonlinear boundary conditions. We study identifiability and local Lipschitz stability. In particular, we prove that the nonlinear term is identified by one Cauchy data set. Moreover, we produce approximated solutions by means of an optimizational method.

We study the inverse problem of the determination of the most favored relative orientations of moving protein domains from Residual Dipolar Coupling (RDC) measurements. We present a numerical procedure based on the simplex method for the efficient determination of the maximum probability of a given relative orientation. We prove the convergence of the algorithm and present the results obtained both on synthetic and on experimental data.