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On the short time asymptotic of the stochastic Allen–Cahn equation

Hendrik Weber (2010)

Annales de l'I.H.P. Probabilités et statistiques

A description of the short time behavior of solutions of the Allen–Cahn equation with a smoothened additive noise is presented. The key result is that in the sharp interface limit solutions move according to motion by mean curvature with an additional stochastic forcing. This extends a similar result of Funaki [Acta Math. Sin (Engl. Ser.)15 (1999) 407–438] in spatial dimension n=2 to arbitrary dimensions.

Optimal measures for the fundamental gap of Schrödinger operators

Nicolas Varchon (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We study the potential which minimizes the fundamental gap of the Schrödinger operator under the total mass constraint. We consider the relaxed potential and prove a regularity result for the optimal one, we also give a description of it. A consequence of this result is the existence of an optimal potential under L1 constraints.

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