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In this paper we first review the covering space method with constrained BV functions for solving the classical Plateau's problem. Next, we carefully analyze some interesting examples of soap films compatible with the covering space method: in particular, the case of a soap film only partially wetting a space curve, a soap film spanning a cubical frame but having a large tunnel, aa soap film that retracts onto its boundary, hence not modelable with the Reifenberg method, and various soap films spanning an octahedral frame.

In this paper, we study the diffusion of a water release into an important real water basin, Lake Garda in northern Italy, coupling a variety of models from hydrodynamics (full Navier-Stokes for the motion of water) to statistics (for the modelling of rain inflow and the time-periodic wind motion on the surface). Using a recent digital bathymetry, we are able to present some novel forecasts of water dynamics, in the presence of an uncommon water release by river Adige into Lake Garda, necessary to prevent flood situations in case of heavy rainfall, and taking into account common inflows, wind speed on the surface and friction on the bottom. The output shows the trajectories of the released particles, which are typically much colder than the existing ones, and may help to individuate critical sites and shores.

There is a need of consensus about the pulmonary rehabilitation (PR) in patients with COVID-19 after discharge from acute care. To facilitate the knowledge of the evidence and its translation into practice, we developed suggestions based on experts’ opinion. A steering committee identified areas and questions sent to experts. Other international experts participated to a RAND Delphi method in reaching consensus and proposing further suggestions. Strong agreement in suggestions was defined when the mean agreement was >7 (1 = no agreement and 9 = maximal agreement). Panelists response rate was >95%. Twenty-three questions from 4 areas: personnel protection equipment, phenotypes, assessments, interventions, were identified and experts answered with 121 suggestions, 119 of which received high level of concordance. The evidence-based suggestions provide the clinicians with current evidence and clinical experts opinion. This framework can be used to facilitate clinical decision making within the context of the individual patient. Further studies will evaluate the clinical usefulness of these suggestions.

Using a suitable triple covering space it is possible to model the construction of a non-simply connected minimal surface spanning all six edges of an elongated tetrahedron, working in the space of BV functions and interpreting the film as the boundary of a Caccioppoli set in the covering space. The possibility of using covering spaces for minimal surfaces was first proposed by Brakke. After a question raised by R. Hardt in the late 1980's, it seems common opinion that an area-minimizing surface of this sort does not exist for a regular tetrahedron, although a proof of this fact is still missing. In this paper we show that there exists a non-simply connected surface spanning the boundary of an elongated tetrahedron and having area strictly less than the area of the minimal contractible surface.

Motivated by a variational model concerning the depth of the objects in a picture and the problem of illusory contours, this book investigates one of the central problems of computer vision: the topological and algorithmical reconstruction of a smooth three dimensional scene starting from the visible part of an apparent contour. The authors focus their attention on the manipulation of apparent contours using a finite set of elementary moves, which correspond to consider diffeomorphic three dimensional scenes. A large part of the book is devoted to the algorithmic part, with implementations, experiments, and computed examples. The book is intended also as a user's guide to the software code appcontour, written for the manipulation of apparent contours and their invariants.

In this paper we study the nonconvex anisotropic mean curvature flow of a hypersurface. This corresponds to an anisotropic mean curvature flow where the anisotropy has a nonconvex Frank diagram. The geometric evolution law is therefore forward-backward parabolic in character, hence ill-posed in general. We study a particular regularization of this geometric evolution, obtained with a nonlinear version of the so-called bidomain model. This is described by a degenerate system of two uniformly parabolic equations of reaction-diffusion type, scaled with a positive parameter $\epsilon$. We analyze some properties of the formal limit of solutions of this system as $\epsilon \to 0$, and show its connection with nonconvex mean curvature flow. Several numerical experiments substantiating the formal asymptotic analysis are presented.

Evolution by mean curvature is recently attracting large attention especially when the underlying anisotropic structure degenerates to become crystalline. In such situation new phenomena must be taken into account: the evolution law becomes nonlocal and "hyperbolic" across facets; moreover events like face breaking or bending have to be considered especially in three dimensions. For this reason the ODE approach suggested by J. Taylor long ago cannot be used directly and the required modifications to the algorithm are not clear at the moment. The well known diffused interface approximation for the classical mean curvature flow, which leads to the Allen-Cahn equation, can be applied in this contex, resulting in a bistable reaction-diffusion equation with good convergence properties to the sharp interface evolution. This equation can then be discretized using finite elements in space and forward differences in time. Numerical simulations with the resulting scheme seem to recover the face breaking and face bending phenomena.