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Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian

D. Le Peutrec (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

This article follows the previous works [HeKlNi, HeNi] by Helffer-Klein-Nier and Helffer-Nier about the metastability in reversible diffusion processes via a Witten complex approach. Again, exponentially small eigenvalues of some self-adjoint realization of Δ f , h ( 0 ) = - h 2 Δ + ∇ f ( x ) 2 - h Δ f ( x ) are considered as the small parameter h > 0 tends to 0 . The function f is assumed to be a Morse function on some bounded domain Ω with boundary ∂ Ω . Neumann type boundary conditions are considered. With these boundary conditions, some possible simplifications...

Smooth optimal synthesis for infinite horizon variational problems

Andrei A. Agrachev, Francesca C. Chittaro (2009)

ESAIM: Control, Optimisation and Calculus of Variations

We study Hamiltonian systems which generate extremal flows of regular variational problems on smooth manifolds and demonstrate that negativity of the generalized curvature of such a system implies the existence of a global smooth optimal synthesis for the infinite horizon problem. We also show that in the Euclidean case negativity of the generalized curvature is a consequence of the convexity of the Lagrangian with respect to the pair of arguments. Finally, we give a generic classification for...

Sobolev-Kantorovich Inequalities

Michel Ledoux (2015)

Analysis and Geometry in Metric Spaces

In a recent work, E. Cinti and F. Otto established some new interpolation inequalities in the study of pattern formation, bounding the Lr(μ)-norm of a probability density with respect to the reference measure μ by its Sobolev norm and the Kantorovich-Wasserstein distance to μ. This article emphasizes this family of interpolation inequalities, called Sobolev-Kantorovich inequalities, which may be established in the rather large setting of non-negatively curved (weighted) Riemannian manifolds by means...

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