Singular perturbations and first order PDE on manifolds

Abstract

In this note we present some results concerning the concentration of sequences of first eigenfunctions on the limit sets of a Morse-Smale dynamical system on a compact Riemanniann manifold. More precisely a renormalized sequence of eigenfunctions converges to a measure $μ$ concentrated on the hyperbolic sets of the field. The set of all possible measure turns out to be a sum of a finite Dirac distributions localized at the critical point of the field and absolutely continuous measure with respect to the Lebesgue measure on each limit cycles : the coefficients which appear in the limit measure can be characterized using the concentration theory. In the second part, certain aspects of some first order PDE on manifolds are studied. We study the limit of a sequence solutions of a second order PDE, when a parameter of viscosity tends to zero. Under some explicit assumptions on some vector fields, bounded and differentiable solutions are obtained. We exibit the role played by the limit sets of the dynamical systems and provide in some cases an explicit representation formula.