论文标题
在异质环境中人口动态模型中资源和施泰纳对称的最佳位置
Optimal location of resources and Steiner symmetry in a population dynamics model in heterogeneous environments
论文作者
论文摘要
本文的主题灵感来自\ cite {cc}和\ cite {ccp}。在\ cite {cc}中，作者通过扩散的逻辑方程来研究异质环境中人群的动力学。他们的研究的重要部分是找到足够的条件来保证该物种的生存。从数学上讲，此任务导致加权特征值问题$-ΔU=λmu $在一个有限的平滑域中$ω\ subset \ subset \ mathbb {r}^n $，$ n \ geq 1 $，在同质的dirichlet边界条件下，其中$λ\ in \ mathbb in \ mathbb in \ mathbb {域$ω$代表环境，$ m（x）$称为本地增长率，说明了有利和不利的栖息地所在的位置。然后，\ cite {cc}中的作者考虑一类权重$ m（x）$，对应于环境，在这些环境中，固定了有利和不利的栖息地的总尺寸，但它们的空间布置可以改变；他们确定了他们之间的最佳选择，以使人口生存。
The subject of this paper is inspired by \cite{CC} and \cite{CCP}. In \cite{CC} the authors investigate the dynamics of a population in a heterogeneous environment by means of diffusive logistic equations. An important part of their study consists in finding sufficient conditions which guarantee the survival of the species. Mathematically, this task leads to the weighted eigenvalue problem $-Δu =λm u $ in a bounded smooth domain $Ω\subset \mathbb{R}^N$, $N\geq 1$, under homogeneous Dirichlet boundary conditions, where $λ\in \mathbb{R}$ and $m\in L^\infty(Ω)$. The domain $Ω$ represents the environment and $m(x)$, called the local growth rate, says where the favourable and unfavourable habitats are located. Then, the authors in \cite{CC} consider a class of weights $m(x)$ corresponding to environments where the total sizes of favourable and unfavourable habitats are fixed, but their spatial arrangement is allowed to change; they determine the best choice among them for the population to survive.\\ In our paper we give an alternative proof and develop a refinement of the result above, moreover we prove a Steiner symmetry result.
