论文标题
加权两阶段dirichlet特征值的形状优化
Shape optimization of a weighted two-phase Dirichlet eigenvalue
论文作者
论文摘要
令$ m $为有界函数，而$α$ a非负参数。本文与漂移的拉普拉斯类型操作员的第一个特征值$λ\_α（m）$有关 状况。假设在$ m $上取得了统一和整体界限，我们研究了相对于$ m $，将$λ\_α（m）$最小化的问题。这样的问题与所谓的“两个阶段极端特征值问题”有关，并且自然出现，例如，在人口动态中，它与物种在域中的生存能力有关。我们证明，除非域是球，否则该问题没有“常规”解决方案。然后，我们通过以下情况进行仔细的分析：（1）在涉及问题的均质版本的新方法的帮助下，在所有径向对称的资源分布之间表征解决方案； （2）在更一般的环境中证明，借助于二阶形状衍生物的单调性原理，将资源的集中分布的稳定性结果显着简化了分析。
Let $m$ be a bounded function and $α$ a nonnegative parameter. This article is concerned with the first eigenvalue $λ\_α(m)$ of the drifted Laplacian type operator $\mathcal L\_m$ given by $\mathcal L\_m(u)= -\operatorname{div} \left((1+αm)\nabla u\right)-mu$ on a smooth bounded domain, with Dirichlet boundary conditions. Assuming uniform pointwise and integral bounds on $m$, we investigate the issue of minimizing $λ\_α(m)$ with respect to $m$. Such a problem is related to the so-called "two phase extremal eigenvalue problem" and arises naturally, for instance in population dynamics where it is related to the survival ability of a species in a domain. We prove that unless the domain is a ball, this problem has no "regular" solution. We then provide a careful analysis in the case of a ball by: (1) characterizing the solution among all radially symmetric resources distributions, with the help of a new method involving a homogenized version of the problem; (2) proving in a more general setting, a stability result for the centered distribution of resources with the help of a monotonicity principle for second order shape derivatives which significantly simplifies the analysis.
