复合膜特征值optimization的对称破缺及其它现象
Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes
摘要：
我们考虑以下特征值最优化问题：给定一个有界域$Ω\subset\R^n$和数$α\geq 0$，$A\in [0,|Ω|]$，找到一个区域$A$的子集$D\subsetΩ$，使算子$-Δ+ αχ_D$的第一Dirichlet特征值尽可能小.我们证明了解的存在性，并研究了它们的定性性质。例如，我们表明，对于一些对称域（薄annuli和哑铃与窄柄）的最优解必须具有较少的对称性比$Ω$ ;另一方面，凸$Ω$反射对称性被保存。此外，我们还给出了数值结果并提出了他们提出的一些猜想。
英文摘要：
We consider the following eigenvalue optimization problem: Given a bounded domain $Ω\subset\R^n$ and numbers $α\geq 0$, $A\in [0,|Ω|]$, find a subset $D\subsetΩ$ of area $A$ for which the first Dirichlet eigenvalue of the operator $-Δ+ αχ_D$ is as small as possible. We prove existence of solutions and investigate their qualitative properties. For example, we show that for some symmetric domains (thin annuli and dumbbells with narrow handle) optimal solutions must possess fewer symmetries than $Ω$; on the other hand, for convex $Ω$ reflection symmetries are preserved. Also, we present numerical results and formulate some conjectures suggested by them.
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