Francesco Maddalena

A variational model of irrigation patterns

Irrigation and draining systems, plants and trees together with their root systems, lungs and cardiovascular systems have a common morphology which seems to derive from topological constraints together with energy saving requirements. All of these systems look like spatial trees and succeed in spreading out a fluid from a source onto a volume. The associated morphology is a tree of bifurcating vessels. Their intuitive explanation is that transport energy is saved by using broad vessels as long as possible rather than thin spread out vessels. In this paper, we define a general formalism dealing with irrigation patterns. Related to martingale theory, this formalism permits one to define irrigation trees and their vessels, to give a generic form to their energy, and to show compactness for the irrigation patterns with bounded energy as well as a lower semicontinuity result for the cost functional. As a consequence, we show that a variety of source to volume irrigation problems are well posed.

Synchronic and Asynchronic Descriptions of Irrigation Problems

Abstract 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.

Hölder continuity conditions for the solvability of Dirichlet problems involving functionals with free discontinuities

Abstract The paper deals with the problem of minimizing a free discontinuity functional under Dirichlet boundary conditions. An existence result was known so far for C 1 (∂Ω) boundary data u . We show here that the same result holds for u ∈C 0,μ (∂Ω) if μ> 1 2 and it cannot be extended to cover the case μ= 1 2 . The proof is based on some geometric measure theoretic properties, in part introduced here, which are proved a priori to hold for all the possible minimizers.

Blow-up techniques and regularity near the boundary for free discontinuity problems

Abstract In this paper we exploit some compactness properties of the set of quasiminimizers of a functional with free discontinuities with the aim of obtaining blow-up techniques. These techniques are finally applied to show regularity near the boundary of the minimizers of the Mumford-Shah functional.

Variational models for peeling problems

We study variational models for flexural beams and plates interacting with a rigid substrate through an adhesive layer. The general structure of the minimizers is investigated and some properties characterizing the behavior of the systems in dependence of the load and the material stiffnesses are discussed.

Regularity properties of free discontinuity sets

Abstract This paper is concerned with the problem of estimating the dimension, expected to be N −2, of the singular set of a minimizer of a functional with free discontinuities in N dimensions. The best result already known, namely the ( N −1)-negligibility, is improved here.

A Metric Approach to Elastic Reformations

We study a variational framework to compare shapes, modeled as Radon measures on

Elastic Structures in Adhesion Interaction

{\mathbb{R}}^{N}

Fractality in selfsimilar minimal mass structures

, 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.

Variational Problems for Föppl--von Kármán Plates

We study a variational model describing the interaction of two one-dimensional elastic bodies through an adhesive layer, with the aim of modeling a simplified CFRP structure: e.g., a concrete beam or a medical rehabilitation device glued to a reinforcing polymeric fiber.