Equilibrium shapes of charged droplets and related problems: (mostly) a review

We review some recent results on the equilibrium shapes of charged liquid drops. We show that the natural variational model is ill-posed and how this can be overcome by either restricting the class of competitors or by adding penalizations in the functional. The original contribution of this note is twofold. First, we prove existence of an optimal distribution of charge for a conducting drop subject to an external electric field. Second, we prove that there exists no optimal conducting drop in this setting

Brunn-Minkowski inequality for the 1-Riesz capacity and level set convexity for the 1/2-Laplacian

We prove that that the 1-Riesz capacity satisfi es a Brunn-Minkowski inequality, and that the capacitary function of the 1/2-Laplacian is level set convex.Comment: 9 page

Shape Optimization Problems for Metric Graphs

We consider the shape optimization problem $$\min\big\{{\mathcal E}(\Gamma)\ :\ \Gamma\in{\mathcal A},\ {\mathcal H}^1(\Gamma)=l\ \big\},$$ where ${\mathcal H}^1$ is the one-dimensional Hausdorff measure and ${\mathcal A}$ is an admissible class of one-dimensional sets connecting some prescribed set of points ${\mathcal D}=\{D_1,\dots,D_k\}\subset{\mathbb R}^d$. The cost functional ${\mathcal E}(\Gamma)$ is the Dirichlet energy of $\Gamma$ defined through the Sobolev functions on $\Gamma$ vanishing on the points $D_i$. We analyze the existence of a solution in both the families of connected sets and of metric graphs. At the end, several explicit examples are discussed.Comment: 23 pages, 11 figures, ESAIM Control Optim. Calc. Var., (to appear

A note on the hausdorff dimension of the singular set for minimizers of the mumford-shah energy

We give a more elementary proof of a result by Ambrosio, Fusco and Hutchinson to estimate the Hausdorff dimension of the singular set of minimizers of the Mumford-Shah energy (see [2, Theorem 5.6]). On the one hand, we follow the strategy of the above mentioned paper; but on the other hand our analysis greatly simplifies the argument since it relies on the compactness result proved by the first two Authors in [4, Theorem 13] for sequences of local minimizers with vanishing gradient energy, and the regularity theory of minimal Caccioppoli partitions, rather than on the corresponding results for Almgren's area minimizing sets

Numerical and Experimental Study on the Friction of Complex Surfaces

Whenever two bodies are in contact due to a normal load and one is sliding against the other, a tangential force arises, as opposed to the motion. This force is called friction force and involves different mechanisms, such as asperity interactions, energy dissipation, chemical and physical alterations of the surface topography and wear. The friction coefficient is defined as the ratio between the friction force and the applied normal load. Despite this apparently simple definition, friction appears to be a very complex phenomenon, which also involves several aspects at both the micro- and nano-scale, including adhesion and phase transformation. Moreover, it plays a key role in a variety of systems, and must be either enhanced (e.g. for locomotion) or minimized (e.g. in bearings), depending on the application. Considering friction as a multiscale problem, an analytical model has been proposed, starting from the literature, to describe friction in the presence of anisotropy, adhesion and wear between surfaces with hierarchical structures, e.g. self-similar. This model has been implemented in a MATLAB code for the design of the tribological properties of hierarchical surfaces and has been applied to study the ice friction, comparing analytical predictions with experimental tests. Furthermore, particular isotropic or anisotropic surface morphologies (e.g., microholes of different shapes and sizes) has been investigated for their influence to the static and dynamic friction coefficients with respect to a flat counterpart. In particular, it has been proved that the presence of grooves on surfaces could decrease the friction coefficients and thus reduce wear and energy dissipation. Experimental tests were performed with a setup realized ad hoc and the results were compared with full numerical simulations. If patterned surfaces showed that they can reduce sliding friction, other applications could require an increase in energy dissipation, e.g. to enhance the toughness of microfibers. Specifically, the applied method consists of introducing sliding frictional elements (sliding knots) in biological (silkworm silk, natural or degummed) and synthetic fibres, reproducing the concept of molecules, where the sacrificial bonds provide higher toughness to the molecular backbone, with a hidden length, which occurs after their breakage. A variety of slip knot topologies with different unfastening mechanisms have been investigated, including even complex knots usually adopted in the textile industry. The knots were made by manipulation of fibres with tweezers and the resulting knotted fibres were characterized through nanotensile tests to obtain their stress-strain curve until failure. The presence of sliding knots strongly increases the dissipated energy per unit mass, without compromising the structural integrity of the fibre itself