Singular perturbation for the first eigenfunction and blow up analysis
| dc.creator | Holcman, David | |
| dc.creator | Kupka, Ivan | |
| dc.date | 2004-12-25 | |
| dc.date.accessioned | 2026-07-07T04:31:46Z | |
| dc.date.available | 2026-07-07T04:31:46Z | |
| dc.description | On a compact Riemannian manifold (V_{m},g), we consider the second order positive operator L_ε = εΔ_{g} +(b,\nabla) +c, where -Δ_{g} is the Laplace-Beltrami operator and b is a Morse-Smale (MS) field, εa small parameter. We study the measures which are the limits of the normalized first eigenfunctions of L_ε as εgoes 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. | |
| dc.description | 70 pages | |
| dc.identifier | https://arxiv.org/abs/math-ph/0412088 | |
| dc.identifier | http://arxiv.org/abs/math-ph/0412088 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/57940 | |
| dc.subject | Mathematical Physics | |
| dc.subject | 58J05 | |
| dc.title | Singular perturbation for the first eigenfunction and blow up analysis | |
| dc.type | text |