Triple covers and a non-simply connected surface spanning an elongated tetrahedron and beating the cone

By using a suitable triple cover we show how to possibly model the construction of a minimal surface with positive genus spanning all six edges of a tetrahedron, working in the space of BV functions and interpreting the film as the boundary of a Caccioppoli set in the covering space. 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 surface of positive genus spanning the boundary of an elongated tetrahedron and having area strictly less than the area of the conic surface.Comment: Expanding on the previous version with additional lower bounds, new images, corrections and improvements. Comparison with Reifenberg approac

Позднеэоценовая регрессия как фактор геотермического режима нефтематеринских отложений арктических районов Западной Сибири (на примере Южного Ямала)

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

Shape Reconstruction from Apparent Contours. Theory and Algorithms

4reservedmixedBellettini, Giovanni; Beorchia, V; Paolini, M; Pasquarelli, F.Bellettini, Giovanni; Beorchia, V; Paolini, M; Pasquarelli, F

Numerical simulation of crystalline curvature flow in 3D by interface diffusion

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

Unstable crystalline Wulff shapes in 3D

We investigate the stability of the evolution by anysotropic and crystalline curvature starting from an initial surface equal to the Wulff shape. It is well known that the Wulff shape evolves selfsimilarly according to the law $V=-\kappa_\phi n_\phi$. Here the index $\phi$ refers to the underlying anisotropy described by the Wulff shape, so that $\kappa_\phi$ is the relative mean curvature and $n_\phi$ is the Cahn-Hoffmann conormal vector field. Such selfsimilar evolution is also known to be stable under small perturbations of the initial surface in the isotropic setting (the Wulff shape is a sphere) or in 2D if the underlying anisotropy is symmetric. We show that this evolution is unstable for some specific choices of the Wulff shape both rotationally symmetric and fully crystalline

Triple covers and a non-simply connected surface spanning an elongated tetrahedron and beating the cone

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

Nonconvex mean curvature flow as a formal singular limit of the nonlinear bidomain model

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

Nonconvex mean curvature flow as a formal singular limit of the nonlinear bidomain model

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 \u3f5. We analyze some properties of the formal limit of solutions of this system as \u3f5\u21920+, and show its connection with nonconvex mean curvature flow. Several numerical experiments substantiating the formal asymptotic analysis are presented

Multimodelling simulation of external water inflows into a morainic lake: the Garda lake case

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