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A Global Uniqueness Result for an Evolution Problem Arising in Superconductivity

Edoardo Mainini — 2009

Bollettino dell'Unione Matematica Italiana

We consider an energy functional on measures in ℝ 2 arising in superconductivity as a limit case of the well-known Ginzburg Landau functionals. We study its gradient flow with respect to the Wasserstein metric of probability measures, whose corresponding time evolutive problem can be seen as a mean field model for the evolution of vortex densities. Improving the analysis made in [AS], we obtain a new existence and uniqueness result for the evolution problem.

On convex sets that minimize the average distance

Antoine Lemenant;  Edoardo Mainini — 2012

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we study the compact and convex sets K ⊆ Ω ⊆ ℝ2that minimize ∫ Ω ( , K ) d + λ 1 Vol ( K ) + λ 2 Per ( K ) ∫ Ω dist ( x ,K ) d x + λ 1 Vol ( K ) + λ 2 Per ( K ) for some constantsλ 1 and λ 2, that could possibly be zero. We compute in particular the second order derivative of the functional and use it to exclude...

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