Journal für die reine und angewandte Mathematik (Crelles Journal)
The motion by curvature of networks is the generalization to finite union of curves of the curve ... more The motion by curvature of networks is the generalization to finite union of curves of the curve shortening flow. This evolution has several peculiar features, mainly due to the presence of junctions where the curves meet. In this paper we show that whenever the length of one single curve vanishes and two triple junctions coalesce, then the curvature of the evolving networks remains bounded. This topological singularity is exclusive of the network flow and it can be referred as a “Type-0” singularity, in contrast to the well known “Type-I” and “Type-II” ones of the usual mean curvature flow of smooth curves or hypersurfaces, characterized by the different rates of blow-up of the curvature. As a consequence, we are able to give a complete description of the evolution of tree–like networks till the first singular time, under the assumption that all the “tangents flows” have unit multiplicity. If the lifespan of such solutions is finite, then the curvature of the network remains bounde...
We consider a variational problem involving competition between surface tension and charge repuls... more We consider a variational problem involving competition between surface tension and charge repulsion. We show that, as opposed to the case of weak (short-range) interactions where we proved ill-posedness of the problem in a previous paper, when the repulsion is stronger the perimeter dominates the capacitary term at small scales. In particular we prove existence of minimizers for small charges as well as their regularity. Combining this with the stability of the ball under small $C^{1,γ}$ perturbations, this ultimately leads to the minimality of the ball for small charges. We cover in particular the borderline case of the $1-$capacity where both terms in the energy are of the same order.
Journal für die reine und angewandte Mathematik (Crelles Journal)
The motion by curvature of networks is the generalization to finite union of curves of the curve ... more The motion by curvature of networks is the generalization to finite union of curves of the curve shortening flow. This evolution has several peculiar features, mainly due to the presence of junctions where the curves meet. In this paper we show that whenever the length of one single curve vanishes and two triple junctions coalesce, then the curvature of the evolving networks remains bounded. This topological singularity is exclusive of the network flow and it can be referred as a “Type-0” singularity, in contrast to the well known “Type-I” and “Type-II” ones of the usual mean curvature flow of smooth curves or hypersurfaces, characterized by the different rates of blow-up of the curvature. As a consequence, we are able to give a complete description of the evolution of tree–like networks till the first singular time, under the assumption that all the “tangents flows” have unit multiplicity. If the lifespan of such solutions is finite, then the curvature of the network remains bounde...
We consider a variational problem involving competition between surface tension and charge repuls... more We consider a variational problem involving competition between surface tension and charge repulsion. We show that, as opposed to the case of weak (short-range) interactions where we proved ill-posedness of the problem in a previous paper, when the repulsion is stronger the perimeter dominates the capacitary term at small scales. In particular we prove existence of minimizers for small charges as well as their regularity. Combining this with the stability of the ball under small $C^{1,γ}$ perturbations, this ultimately leads to the minimality of the ball for small charges. We cover in particular the borderline case of the $1-$capacity where both terms in the energy are of the same order.
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Papers by Matteo Novaga