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Alessandra Pluda

Publications and source records attributed to Alessandra Pluda.

27 records · Page 2Linked to original sources
arXiv2019

Calibrations for minimal networks in a covering space setting

In this paper we define a notion of calibration for an equivalent approach to the classical Steiner problem in a covering space setting and we give some explicit examples. Moreover we introduce the notion of calibration in families: the idea is to divide the set of competitors in a suitable way, defining an appropriate (and weaker) notion of calibration. Then, calibrating the candidate minimizers in each family and comparing their perimeter, it is possible to find the minimizers of the minimization problem. Thanks to this procedure we prove the minimality of the Steiner configurations spanning the vertices of a regular hexagon and of a regular pentagon.

math.OC↗
arXiv2019

On different notions of calibrations for minimal partitions and minimal networks in $\mathbb{R}^2$

Calibrations are a possible tool to validate the minimality of a certain candidate. They have been introduced in the context of minimal surfaces and adapted to the case of Steiner problem in several variants. Our goal is to compare the different notions of calibrations for the Steiner Problem and for planar minimal partitions. The paper is then complemented with remarks on the convexification of the problem, on non-existence of calibrations and on calibrations in families.

math.OC↗
arXiv2019

Long Time Existence of Solutions to an Elastic Flow of Networks

The $L^2$--gradient flow of the elastic energy of networks leads to a Willmore type evolution law with nontrivial 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.

math.AP↗
arXiv2017

Willmore Flow of planar networks

Geometric gradient flows for elastic energies of Willmore type play an important role in mathematics and in many applications. The evolution of elastic curves has been studied in detail both for closed as well as for open curves. Although elastic flows for networks also have many interesting features, they have not been studied so far from the point of view of mathematical analysis. So far it was not even clear what are appropriate boundary conditions at junctions. In this paper we give a well-posedness result for Willmore flow of networks in different geometric settings and hence lay a foundation for further mathematical analysis. A main point in the proof is to check whether different proposed boundary conditions lead to a well posed problem. In this context one has to check the Lopatinskii--Shapiro condition in order to apply the Solonnikov theory for linear parabolic systems in Hölder spaces which is needed in a fixed point argument. We also show that the solution we get is unique in a purely geometric sense.

math.AP↗
arXiv2017

Some minimization problems for planar networks of elastic curve

In this note we announce some results that will appear in [6] (joint work with also Matteo Novaga) on the minimization of the functional $F(Γ)=\int_Γk^2+1\,\mathrm{d}s$, where $Γ$ is a network of three curves with fixed equal angles at the two junctions. The informal description of the results is accompanied by a partial review of the theory of elasticae and a diffuse discussion about the onset of interesting variants of the original problem passing from curves to networks. The considered energy functional $F$ is given by the elastic energy and a term that penalize the total length of the network. We will show that penalizing the length is tantamount to fix it. The paper is concluded with the explicit computation of the penalized elastic energy of the 'Figure Eight', namely the unique closed elastica with self--intersections.

math.AP↗
arXiv2016

Evolution of spoon-shaped networks

We consider a regular embedded network composed by two curves, one of them closed, in a convex domain $Ω$. The two curves meet only in one point, forming angle of $120$ degrees. The non-closed curve has a fixed end point on $\partialΩ$. We study the evolution by curvature of this network. We show that the maximal existence time depends only on the area enclosed in the initial loop, if the length of the non-closed curve stays bounded from below during the evolution. Moreover, the closed curve shrinks to a point and the network is asymptotically approaching, after dilations and extraction of a subsequence, a Brakke spoon.

math.AP↗
arXiv2015

Spines of minimal length

In this paper we raise the question whether every closed Riemannian manifold has a spine of minimal area, and we answer it affirmatively in the surface case. On constant curvature surfaces we introduce the spine systole, a continuous real function on moduli space that measures the minimal length of a spine in each surface. We show that the spine systole is a proper function and has its global minima precisely on the extremal surfaces (those containing the biggest possible discs). We also study minimal spines, which are critical points for the length functional. We completely classify minimal spines on flat tori, proving that the number of them is a proper function on moduli space. We also show that the number of minimal spines of uniformly bounded length is finite on hyperbolic surfaces.

math.GT↗