Singular perturbation for the first eigenfunction and blow up analysis

On a compact Riemannian manifold (V_{m},g), we consider the second order positive operator L_{\epsilon} = \epsilon\Delta_{g} +(b,\nabla) +c, where -\Delta_{g} is the Laplace-Beltrami operator and b is a Morse-Smale (MS) field, \epsilon a small parameter. We study the measures which are the limits of the normalized first eigenfunctions of L_{\epsilon} as \epsilon goes to the zero. In the case of a general MS field $b$, such a limit measures is the sum of a linear combination of Dirac measures located at the singular point of $b$ and a linear combination of measures supported by the limit cycles of b. When b is a MS-gradient vector field, we use a Blow-up analysis to determine how the sequence concentrates on the critical point set. We prove that the set of critical points that a critical point belongs to the support of a limit measure only if the Topological Pressure defined by a variational problem is achieved there. Also if a sequence converges to a measure in such a way that every critical points is a limit point of global maxima of the eigenfunction, then we can compute the weight of a limit measure.This result provides a link between the limits of the first eigenvalues and the associated eigenfunctions . We give an interpretation of this result in term of the movement of a Brownian particle driven by a field and subjected to a potential well, in the small noise limit.