论文标题
在复合膜问题中的最佳配置和对称性破坏现象的分数拉普拉斯
Optimal configuration and symmetry breaking phenomena in the composite membrane problem with fractional Laplacian
论文作者
论文摘要
我们在复合膜问题中考虑以下特征值优化的分数laplacian：给定一个有界域$ω\ subset \ subset \ mathbb {r}^n $，$α> 0 $和$ 0 <a <a <|ω| $ $（ - δ）^s+αχ_d$尽可能小。解决方案$ d $称为数据$（ω，α，a）$的最佳配置。查看众所周知的分数laplacian的扩展定义，在$ s = 1/2 $的情况下，这本质上是复合膜问题，质量集中在边界上，因为人们试图最大程度地提高steklov eigenvalue。 我们证明了解决方案的存在和最佳配置$ d $的研究属性。这是一个自由边界问题，可以作为双面不稳定的障碍物问题提出。 此外，我们表明，对于某些旋转的对称域（薄annuli），最佳配置不是旋转对称的，这意味着最佳配置$ d $的非唯一性。另一方面，我们证明，对于具有反射对称性的凸域$ω$，最佳配置具有相同的对称性，这意味着球盒中最佳配置$ d $的唯一性。
We consider the following eigenvalue optimization in the composite membrane problem with fractional Laplacian: given a bounded domain $Ω\subset \mathbb{R}^n$, $α>0$ and $0<A<|Ω|$, find a subset $D\subset Ω$ of area $A$ such that the first Dirichlet eigenvalue of the operator $(-Δ)^s+αχ_D$ is as small as possible. The solution $D$ is called as an optimal configuration for the data $(Ω,α,A)$. Looking at the well-known extension definition for the fractional Laplacian, in the case $s=1/2$ this is essentially the composite membrane problem for which the mass is concentrated at the boundary as one is trying to maximize the Steklov eigenvalue. We prove existence of solutions and study properties of optimal configuration $D$. This is a free boundary problem which could be formulated as a two-sided unstable obstacle problem. Moreover, we show that for some rotationally symmetric domains (thin annuli), the optimal configuration is not rotational symmetric, which implies the non-uniqueness of the optimal configuration $D$. On the other hand, we prove that for a convex domain $Ω$ having reflection symmetries, the optimal configuration possesses the same symmetries, which implies uniqueness of the optimal configuration $D$ in the ball case.
