David Holcman, Ivan Kupka
Singular perturbation for the first eigenfunction and blow-up analysis
On a compact Riemannian manifold (Vm, g), we consider the second order positive operator
, where − Δb is the Laplace-Beltrami operator and b is a Morse-Smale (MS) field, ϵ a small parameter. We study the measures which are the limits of the normalized first eigenfunctions of Lϵ as ϵ goes to zero.
In the case of a general MS field b, such a limit measure 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 an MS-gradient vector field, we use a blow-up analysis to determine how the sequence concentrates on the critical point set. We prove that a critical point belongs to the support of a limit measure only if the Topological Pressure defined by a variational problem (see [Kifer Y.: Principal eigenvalues, topological pressure, and stochastic stability of equilibrium states. Israel J. Math. 70 (1990), 1–47]) 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 terms of the movement of a Brownian particle driven by a field and subjected to a potential well, in the small noise limit.
Forum Mathematicum, Walter de Gruyter
Print ISSN: 0933-7741
Volume: 18, 05/2006
Pages: 445 - 518
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