1,685 research outputs found
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
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
Shape Optimization Problems for Metric Graphs
We consider the shape optimization problem where is the one-dimensional Hausdorff measure and is an
admissible class of one-dimensional sets connecting some prescribed set of
points . The cost
functional is the Dirichlet energy of defined
through the Sobolev functions on vanishing on the points . 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
Networking Networkers: An Initial Exploration of the Patterns of Collaboration among the Members of a New Community in Political Science
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