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 Riemannian manifold. More precisely a renormalized sequence of eigenfunctions converges to a measure μ concentrated on the hyperbolic sets of the field. 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 of solutions of a second order PDE, when a parameter of viscosity tends to zero. We exhibit the role played by the limit sets of the vector fields.