譜分數型橢圓算子之特徵值單調性

本篇論文以分數階橢圓算子在狄利克雷條件下的權函數(weight function)特徵值問題為出發點，探討在權函數固定的狀況下值域變動所得出之特徵值變化。對應於有界值域變動的狄利克雷條件拉普拉斯算子的特徵值單調性已被證明，我們將會證明分數階橢圓算子也有相同結果。此外，我們也會證明若將此特徵值問題的主要特徵值試為對值域的函數，則其為連續函數。在證明上我們將會運用對應分數階橢圓算子的瑞利商(Rayleigh quotient)及一些相關的最大值—最小值定理。

關鍵字

分數型橢圓算子；分數型索伯列夫空間；特徵值問題；單調性；瑞利商；狄利克雷邊界條件

並列摘要

The eigenvalue problem for the classical elliptic operator, for example, the Laplace operator $-\Delta$, was well-studied. It is well-known that the eigenvalue of the Dirichlet Laplacian is monotone with respect to the domain. In this thesis, we will generalize the ideas to the spectral fractional elliptic operators. We consider the weighted eigenvalue problem involving the spectral fractional elliptic operator $L^{\gamma}$ for $\gamma \in (0,1)$ on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^{n}$. Here we remark that the operator $L^{\gamma}$ is well-defined for all $\gamma \in \mathbb{R}\setminus\mathbb{N}$. Similar to the classical case, we want to show that the eigenvalue of $L^{\gamma}$ is monotone with respect to the domain. Moreover, the principal eigenvalue is continuous when it is viewed as a function of domains. For our purpose, we will establish a corresponding Rayleigh quotient. Last but not least, we remark that the domain monotonicity of the eigenvalues does not hold for Neumann Laplacian. We will elaborate a counter-example in [13] and [14].

並列關鍵字

fractional elliptic operator ； fractional Sobolev space ； eigenvalue problem ； monotonicity property ； Rayleigh quotient ； Dirichlet boundary condition