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Characterization of optimal shapes and masses through Monge-Kantorovich equation

Guy BouchittéGiuseppe Buttazzo — 2001

Journal of the European Mathematical Society

We study some problems of optimal distribution of masses, and we show that they can be characterized by a suitable Monge-Kantorovich equation. In the case of scalar state functions, we show the equivalence with a mass transport problem, emphasizing its geometrical approach through geodesics. The case of elasticity, where the state function is vector valued, is also considered. In both cases some examples are presented.

On the curvature and torsion effects in one dimensional waveguides

Guy BouchittéM. Luísa MascarenhasLuís Trabucho — 2007

ESAIM: Control, Optimisation and Calculus of Variations

We consider the Laplace operator in a thin tube of 3 with a Dirichlet condition on its boundary. We study asymptotically the spectrum of such an operator as the thickness of the tube's cross section goes to zero. In particular we analyse how the energy levels depend simultaneously on the curvature of the tube's central axis and on the rotation of the cross section with respect to the Frenet frame. The main argument is a -convergence theorem for a suitable sequence of quadratic energies.

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