Discrete dynamics of complex bodies with substructural dissipation: Variational integrators and convergence
Matteo Focardi Paolo Maria Mariano
For the linearized setting of the dynamics of complex bodies we construct variational integrators and prove their convergence by making use of BV estimates on the rate fields. We allow for peculiar substructural inertia and internal dissipation, all accounted for by a d'Alembert-Lagrange-type principle.
keywords: dynamical systems asynchronous variational integrators convergence Complex bodies
On a 1-capacitary type problem in the plane
Matteo Focardi Maria Stella Gelli Giovanni Pisante
We study a $1$-capacitary type problem in $R^2$: given a set $E$, we minimize the perimeter (in the sense of De Giorgi) among all the sets containing $E$ (modulo $H^1$) and satisfying an indecomposability constraint (according to the definition by [1]. By suitably choosing the representant of the relevant set $E$, we show that a convexification process characterizes the minimizers.
    As a consequence of our result we determine the $1$-capacity of (a suitable representant of) sets with finite perimeter in the plane.
keywords: capacity Perimeter indecomposable sets.
Vector-valued obstacle problems for non-local energies
Matteo Focardi
We investigate the asymptotics of obstacle problems for non-local energies in a vector-valued setting. Motivations arise, in particular, in phase field models for ferroelectric materials and variational theories for dislocations.
keywords: $\Gamma$-convergence. non-local energies Obstacle problems Delone sets of points
Matteo Focardi Paolo Maria Mariano
Human efforts of describing the mechanisms of physical world are a continuous source of mathematical problems. Models are, in fact, representations of classes of phenomena inspired and corroborated by observations intended here as interpretative cataloguing of events. Mathematics is a language having both qualitative and quantitative nature -- the only one with both features -- so it is natural to use it in constructing representations with predictive characteristics of the events in the phenomenological world. In any other human language, in fact, when we refer to quantification, we use concepts coming from mathematics -- the standard use of numbers in counting, for example, the words in this paragraph is a quantitative feature based on a mathematical concept (the one of numbers indeed) that is added to any type of evaluation of the literary/philosophical quality of the phrases themselves. Models appear then as mathematical structures with constituents constrained by the need of having a clear physical meaning. The mathematical questions appearing in models of the physical world have twofold nature: On one side we have technical hitches related, for example, to existence and regularity of solutions to variational minimality requirements and/or to balance equations under boundary and/or initial contitions. On the other side there are foundational problems generated by the quest of the most appropriate form of models. The target is in fact the physical world: appropriateness is then, for a model, the ability to describe the essential structures underlying some classes of phenomena, and their interconnections in non-trivial way, remaining at the same time rather flexible to allow one the possibility of hopefully describing unexpected features that the experimental programs could put in evidence. A model, then, has to be considered not as a manner of justifying the development of a more or less difficult exercise in mathematics, rather it is an occasion of exploring by the tools of a language both qualitative and quantitative, as mathematics is, the intricate and elegant aspects of the physical world. Distinguishing between different possible models of the same class of phenomena is a volatile matter decided by sensibility and culture (the order is not accidental) of the researcher overburden with the judgement.

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