The quality assessment of manufacturing operations performed to obtain given flat surfaces is always a problem of comparing the substitute model (approximating the features of the true manufactured part) to the nominal specifications, at any stage of the manufacturing cycle. A novel methodology, based on applications of classical tools of Calculus of Variations, is here presented with the aim of assessing the output quality of manufactured flat surfaces based on the information available on transformation imposed by technological processes. By assuming that any manufacturing process operates under equilibrium states, the proposed variational methodology allows to account for the traces left by different stages of manufacturing processes. A simple two-dimensional case is here discussed, to give the flavor of the methodology and its future potential developments.

We derive a non-linear differential equation that must be satisfied by the quantum potential, in the context of the Madelung equations, in one dimension for a particular class of wave functions. In this case, we exhibit explicit conditions leading to the blow-up of the quantum potential of a free particle at the boundary of the compact support of the probability density.

We study a variational framework to compare shapes, modeled as Radon measures on ℝN, in order to quantify how they differ from isometric copies. To this purpose we discuss some notions of weak deformations termed reformations as well as integral functionals having some kind of isometries as minimizers. The approach pursued is based on the notion of pointwise Lipschitz constant leading to a matric space framework. In particular, to compare general shapes, we study this reformation problem by using the notion of transport plan and Wasserstein distances as in optimal mass transportation theory.

We study variational problems modeling the adhesion interaction with a rigid substrate for elastic strings and rods. We produce conditions characterizing bonded and detached states as well as optimality properties with respect to loading and geometry. We show Euler equations for minimizers of the total energy outside self-contact and secondary contact points with the substrate.

The reformation of a body fundamentally involves the mapping of one natural reference configuration of it into another natural reference configuration. The mass and the constitutive properties of the material remain unaltered, but the overall shape of the reference configuration generally changes. If, when a natural reference configuration is distorted, there is a portion of the boundary of the body that is displacement controlled, then a reformation of the body must be such that the original displacement controlled part of the boundary and its reformation are identical. In common applications that involve reformation, the remainder of the boundary is traction-free and a reformation essentially involves a change of the morphology of this traction-free surface. For example, undulations are often a characteristic feature of the reformation of a free, plane boundary surface. Reformations are a result of a material instability and they may associate with a chemically induced diffusive processes in which particles of the body move into preferred places. Fundamentally, a reformation is generated in response to the drive to lower the total stored energy of the body. In this work we are not concerned with the physical processes that take place during reformation, but rather we are concerned with characterizing the onset of the instability. We develop a variational characterization of the reformation instability for a nonlinear elastic body and we include the effect of surface energy. As an example, we consider the axial deformation of a circular cylinder and argue that small scale nano-wires, for which the diameter-to-length ratio is sufficiently small, are expected to be stable with respect to spatial variations when extended. Moreover, we observe that if the surfacial energy function is sufficiently convex at the undistorted state such wires may also be stable with respect to spatial variations when compressed. We then show that such small scale nano-wires are unstable with respect to reformation when either extended or compressed. © 2011 Springer Science+Business Media B.V.

In this paper we complete our work started in [31], where the present paper was announced; in order to set a unified theory of the irrigation problem. The main result of the paper is the equivalence of the various formulations introduced so far as well as a new one introduced here. To this aim we introduce several geometric and analytical concepts which are essential for reaching our final goal even if they may deserve an intrinsic interest in themselves.

By disintegration of transport plans it is introduced the notion of transport class. This allows to consider the Monge problem as a particular case of the Kantorovich transport problem, once a transport class is fixed. The transport problem constrained to a fixed transport class is equivalent to an abstract Monge problem over a Wasserstein space of probability measures. Concerning solvability of this kind of constrained problems, it turns out that in some sense the Monge problem corresponds to a lucky case.