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Singular perturbation for the first eigenfunction and blow up analysis
Mathematical Physics · Submitted 25 Dec 2004
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | background 0.87 | 0.55 | 0.03 | asserted 0.85 |
| s1 | We study the measures which are the limits of the normalized first eigenfunctions of L_{\epsilon} as \epsilon goes to the zero. | contribution 0.99 | 0.91 | 0.08 | asserted 0.91 |
| s2 | 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. | result 1.00 | 0.80 | 0.45 | asserted 1.00 |
| s3 | 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. | method 0.93 | 0.93 | 0.09 | asserted 0.99 |
| s4 | 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. | result 0.99 | 0.91 | 0.33 | asserted 1.00 |
| s5 | 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 . | result 0.99 | 0.92 | 0.22 | asserted 0.89 |
| s6 | 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. | interpretation 1.00 | 0.84 | 0.08 | asserted 0.98 |
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Original abstract
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.
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Papers whose main result relates to this one's
States the same result 1 paper
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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.
Singular perturbations and first order PDE on manifolds · 0.89
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