一类带非局部边界条件的二阶椭圆算子族的谱分析
Spectral Analysis of a Family of Second-Order Elliptic Operators with Nonlocal Boundary Condition Indexed by a Probabilty Measure
摘要：
设$D\subset R^d$是一个有界区域，\[ L=\frac12\nabla\cdot a\nabla +b\cdot\nabla \] % \[ %L=\frac12\sum_{i,j=1}^da_{i,j}\frac{\partial^2}{\partial x_i\partial x_j}+\sum_{i=1}^db_i\frac{\partial}{\partial x_i}, %\]是$D$上的一个二阶椭圆算子.假设$ν$是$D$上的概率测度。用${\mathcal L}$表示微分算子，其区域由以下非局部边界条件：$$ {\mathcal D_{\mathcal L}}=\{f\in C^2(\ol{D}): \int_D f dν= f|_{\partial D}\}, $$指定，并且在其区域上与$L$一致。已知$\mathcal L$具有无穷特征值序列，并且除了零特征值之外，所有特征值都具有负实部。定义$\mathcal {L}$的光谱间隙，由$ν$索引，由γ_1（ν）\equiv \sup \{ \re λ：0 \neq λ是{ \mathcal L}\}的本征值。在本文中，我们调查的特征值$\mathcal L$一般和频谱间隙$γ_1(ν)$特别。算子$\mathcal L$是从边界随机跳跃的扩散过程的生成元，并且$γ_1(ν)$测量该过程收敛到其不变测度的指数速率。
英文摘要：
Let $D\subset R^d$ be a bounded domain and let \[ L=\frac12\nabla\cdot a\nabla +b\cdot\nabla \] %\[ %L=\frac12\sum_{i,j=1}^da_{i,j}\frac{\partial^2}{\partial x_i\partial x_j}+\sum_{i=1}^db_i\frac{\partial}{\partial x_i}, %\] be a second order elliptic operator on $D$. Let $ν$ be a probability measure on $D$. Denote by ${\mathcal L}$ the differential operator whose domain is specified by the following non-local boundary condition: $$ {\mathcal D_{\mathcal L}}=\{f\in C^2(\ol{D}): \int_D f dν= f|_{\partial D}\}, $$ and which coincides with $L$ on its domain. It is known that $\mathcal L$ possesses an infinite sequence of eigenvalues, and that with the exception of the zero eigenvalue, all eigenvalues have negative real part. Define the spectral gap of $\mathcal {L}$, indexed by $ν$, by γ_1(ν)\equiv\sup\{\re λ:0\neq λis an eigenvalue for {\mathcal L}\}. In this paper we investigate the eigenvalues of $\mathcal L$ in general and the spectral gap $γ_1(ν)$ in particular. The operator $\mathcal L$ is the generator of a diffusion process with random jumps from the boundary, and $γ_1(ν)$ measures the exponential rate of convergence of this process to its invariant measure.
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