第一特征函数的奇异摄动与爆破分析
Singular perturbation for the first eigenfunction and blow up analysis
摘要：
在紧致黎曼流形（V_{m}，g）上，考虑二阶正算子L_ε = εΔ_{g} +（b，\nabla）+c，其中-Δ_{g}为Laplace-Beltrami算子，b为Morse-Smale（MS）域，ε为小参数.研究了L_ε的归一化第一本征函数在ε趋于零时的极限测度。在一般MS域$b$的情况下，这样的极限测度是位于$b$的奇点处的狄拉克测度的线性组合与由b的极限环支持的测度的线性组合的和。当b是MS-梯度向量场时，我们使用Blow-up分析来确定序列如何集中在临界点集上。证明了临界点属于极限测度的支撑的临界点集仅当变分问题定义的拓扑压力在那里达到。如果一个序列收敛到一个测度，使得每一个临界点都是特征函数全局最大值的极限点，那么我们可以计算极限测度的权，这个结果提供了第一特征值的极限和相关特征函数的极限之间的联系。在小噪声极限下，我们用布朗粒子在势阱中受场驱动的运动来解释这一结果。
英文摘要：
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.
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