论文标题
球形分层介质中传输本征的边界定位
Boundary localization of transmission eigenfunctions in spherically stratified media
论文作者
论文摘要
考虑$ u \在h^1（ω）$中的传输特征值问题，而与$（ω;σ，\ MathBf {n}^2）$相关的H^1（ω）$中的$ v \ in $ v \ in $（ω;σ，\ Mathbf {n}^2）$，其中$ω$是$ \ \ \ \ \ \ \ m}^r}^n $ n $ n $ n = 2,3 $ in $ \ mathbb {r}^n $ n = 2,3 $。如果$σ$和$ \ mathbf {n} $都是径向对称的，即仅是径向参数$ r $的函数，我们表明存在一系列传输eigenfunctions $ \ \ {u_m，v_m \} $ m \ rightArrow+\ infty $，使$ l^2 $ eenergies的$ v_m $ $集中在$ \partialΩ$附近。如果$σ$和$ \ Mathbf {n} $都是恒定的，我们显示了传输eigenfunctions $ \ {u_j，v_j \} _ {我们的结果扩展了[15,16]的最新研究。通过Numerics，我们还讨论了中等参数的影响，即$σ$和$ \ Mathbf {n} $，对传输特征函数的几何模式。
Consider the transmission eigenvalue problem for $u \in H^1(Ω)$ and $v\in H^1(Ω)$ associated with $(Ω; σ, \mathbf{n}^2)$, where $Ω$ is a ball in $\mathbb{R}^N$, $N=2,3$. If $σ$ and $\mathbf{n}$ are both radially symmetric, namely they are functions of the radial parameter $r$ only, we show that there exists a sequence of transmission eigenfunctions $\{u_m, v_m\}_{m\in\mathbb{N}}$ associated with $k_m\rightarrow+\infty$ as $m\rightarrow+\infty$ such that the $L^2$-energies of $v_m$'s are concentrated around $\partialΩ$. If $σ$ and $\mathbf{n}$ are both constant, we show the existence of transmission eigenfunctions $\{u_j, v_j\}_{j\in\mathbb{N}}$ such that both $u_j$ and $v_j$ are localized around $\partialΩ$. Our results extend the recent studies in [15,16]. Through numerics, we also discuss the effects of the medium parameters, namely $σ$ and $\mathbf{n}$, on the geometric patterns of the transmission eigenfunctions.
