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Liouvillian first integrals of differential equations

Guy Casale (2011)

Banach Center Publications

In this paper we generalize to any dimension and codimension some theorems about existence of Liouvillian solutions or first integrals proved by M. Singer in Liouvillian first integrals of differential equations (1992) for first order differential equations.

Lipschitzian norm estimate of one-dimensional Poisson equations and applications

Hacene Djellout, Liming Wu (2011)

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

By direct calculus we identify explicitly the lipschitzian norm of the solution of the Poisson equation in terms of various norms of g, where is a Sturm–Liouville operator or generator of a non-singular diffusion in an interval. This allows us to obtain the best constant in the L1-Poincaré inequality (a little stronger than the Cheeger isoperimetric inequality) and some sharp transportation–information inequalities and concentration inequalities for empirical means. We conclude with several illustrative...

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