论文标题
具有球形重量和差点边界条件的最佳主特征值的渐近特性
Asymptotic properties of an optimal principal eigenvalue with spherical weight and Dirichlet boundary conditions
论文作者
论文摘要
We consider a weighted eigenvalue problem for the Dirichlet laplacian in a smooth bounded domain $Ω\subset \mathbb{R}^N$, where the bang-bang weight equals a positive constant $\overline{m}$ on a ball $B\subsetΩ$ and a negative constant $-\underline{m}$ on $Ω\setminus B$.相应的正主特征值提供了一个阈值，以检测一种物种的持久性/灭绝，该物种的进化是由异质性的Fisher-kpp方程在种群动力学中描述的。特别是，我们研究了这种特征值在$ω$中的位置的最小化。我们在奇异的扰动方案中提供了最佳特征帕尔的急剧渐近扩展，其中$ b $消失了。我们将其推断为最佳球以最大化$ \partialΩ$的距离。
We consider a weighted eigenvalue problem for the Dirichlet laplacian in a smooth bounded domain $Ω\subset \mathbb{R}^N$, where the bang-bang weight equals a positive constant $\overline{m}$ on a ball $B\subsetΩ$ and a negative constant $-\underline{m}$ on $Ω\setminus B$. The corresponding positive principal eigenvalue provides a threshold to detect persistence/extinction of a species whose evolution is described by the heterogeneous Fisher-KPP equation in population dynamics. In particular, we study the minimization of such eigenvalue with respect to the position of $B$ in $Ω$. We provide sharp asymptotic expansions of the optimal eigenpair in the singularly perturbed regime in which the volume of $B$ vanishes. We deduce that, up to subsequences, the optimal ball concentrates at a point maximizing the distance from $\partialΩ$.
