双连通平面区域的极值第一Dirichlet eigenvalue和二面角对称性
Extremal first Dirichlet eigenvalue of doubly connected plane domains and dihedral symmetry
摘要：
我们处理如下特征值优化问题：给定一个有界区域$D\subset \R^2$，如何在$D$内放置一个固定形状的障碍$B$，以使$D\setminus B$上Dirichlet Laplacian的基本特征值$λ_1$最大或最小.这意味着我们要使函数$ρ\mapsto λ_1(D\setminus ρ(B))$极值化，其中$ρ$在刚性运动集合上运行，使得$ρ(B)\subset D$。我们回答这个问题的情况下，$D$和$B$是不变的作用下的二面角群$\mathbb{D}_n$，$n\ge2$，从原点到边界的距离是单调的两个对称轴之间的参数的函数。极端构型对应于$B$的对称轴与$D$的对称轴重合的情况。
英文摘要：
We deal with the following eigenvalue optimization problem: Given a bounded domain $D\subset \R^2$, how to place an obstacle $B$ of fixed shape within $D$ so as to maximize or minimize the fundamental eigenvalue $λ_1$ of the Dirichlet Laplacian on $D\setminus B$. This means that we want to extremize the function $ρ\mapsto λ_1(D\setminus ρ(B))$, where $ρ$ runs over the set of rigid motions such that $ρ(B)\subset D$. We answer this problem in the case where both $D$ and $B$ are invariant under the action of a dihedral group $\mathbb{D}_n$, $n\ge2$, and where the distance from the origin to the boundary is monotonous as a function of the argument between two axes of symmetry. The extremal configurations correspond to the cases where the axes of symmetry of $B$ coincide with those of $D$.
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