Lectures on curvature flow of networks

We present a collection of results on the evolution by curvature of networks of planar curves. We discuss in particular the existence of a solution and the analysis of singularities

Degenerate Elastic Networks

We minimize a linear combination of the length and the L2-norm of the curvature among networks in Rd belonging to a given class determined by the number of curves, the order of the junctions, and the angles between curves at the junctions. Since this class lacks compactness, we characterize the set of limits of sequences of networks bounded in energy, providing an explicit representation of the relaxed problem. This is expressed in terms of the new notion of degenerate elastic networks that, rather surprisingly, involves only the properties of the given class, without reference to the curvature. In the case of d= 2 we also give an equivalent description of degenerate elastic networks by means of a combinatorial definition easy to validate by a finite algorithm. Moreover we provide examples, counterexamples, and additional results that motivate our study and show the sharpness of our characterization

Long time existence of solutions to an elastic flow of networks

The L2-gradient flow of the elastic energy of networks leads to a Willmore type evolution law with non-trivial nonlinear boundary conditions. We show local in time existence and uniqueness for this elastic flow of networks in a Sobolev space setting under natural boundary conditions. In addition, we show a regularisation property and geometric existence and uniqueness. The main result is a long time existence result using energy methods

Inhaler technique knowledge and skills before and after an educational program in obstructive respiratory disease patients: A real-life pilot study

Patients present poor knowledge and skills about their respiratory disease and inhaler device. We aimed to: (1) evaluate COPD and asthmatic patients' ability to manage inhaled drugs (2) identify differences among devices and (3) correlate clinical data with patient ability.Patients (n=134) admitted for pulmonary rehabilitation (PR) were given an ad-hoc questionnaire covering 0% as the worst and 100% the best value of global ability (indicating the sum of knowledge and skills in managing inhaled drugs) at baseline (T0) and discharge (T1). Educational program was provided during PR. Setting of rehabilitation, age, sex, diagnosis, spirometry, CIRS score, level of autonomy to use medications, if naïve about PR, educational level, and number/type of prescribed inhaled drugs were recorded.Most patients used 1 drug while 37% used 2 drugs. DPIs were the main device prescribed. At baseline, patients' mean level of knowledge and skills were 73% and 58%, respectively. There was a significant difference in level of skills (p=0.046) among device families, DPIs resulting worst and pMDIs best. Global ability, skills and knowledge improved after educational support (p0.001) but did not reach the optimal level, 88%, 87% and 89%, respectively. Baseline global ability was positively correlated to female gender, younger age, previous PR access, outpatient status, higher education level and GOLD D class.At hospital admission, global ability was not optimal. Education may improve this, irrespective of the type of device used, in particular in male, elderly, naïve to PR, low educational level patients

Existence and uniqueness of the motion by curvature of regular networks

We prove existence and uniqueness of the motion by curvature of networks with triple junctions in Rd when the initial datum is of class Wp2-2/p and the unit tangent vectors to the concurring curves form angles of 120 degrees. Moreover, we investigate the regularisation effect due to the parabolic nature of the system. An application of the well-posedness is a new proof and a generalisation of the long-time behaviour result derived by Mantegazza et al. in 2004. Our study is motivated by an open question proposed in the 2016 survey from Mantegazza et al.: does there exist a unique solution of the motion by curvature of networks with initial datum being a regular network of class C 2? We give a positive answer

Variational models for the interaction of surfactants with curvature – existence and regularity of minimizers in the case of flexible curves

Existence and regularity of minimizers for a geometric variational problem is shown. The variational integral models an energy contribution of the interface between two immiscible fluids in the presence of surfactants and includes a Helfrich type contribution, a Frank type contribution and a coupling term between the orientation of the surfactants and the curvature of the interface. Analytical results are proven in a one–dimensional situation for curves

The oriented mailing problem and its convex relaxation

In this note we introduce a new model for the mailing problem in branched transportation that takes into account the orientation of the moving particles. This gives an effective answer to Bernot et al. (2009, Problem 15.9). Moreover we define a convex relaxation in terms of rectifiable currents with group coefficients. We provide the problem with a notion of calibration. Using similar techniques we define a convex relaxation and a corresponding notion of calibration for a variant of the Steiner tree problem in which a connectedness constraint is assigned only among a certain partition of a given set of finitely many points