半圆柱体内一个过定特征值问题的分支域
Bifurcating domains for an overdetermined eigenvalue problem in cylinders
AI总结：
本文研究了半圆柱体内过定特征值问题，通过构造分支域来生成非平凡解。
摘要：
我们研究了包含于半圆柱Σ=ω×(0,+∞)中的域Ω的过定特征值问题，基于一个有界正则域ω⊂R^{N-1}。可以发现，在任何有界圆柱Ω_t=ω×(0,t)，t>0中，特征值问题都存在一个一维正特征函数，满足过定边界条件。本文的目标是构造其他域Ω⊂Σ，使得存在一个正特征函数，是过定问题的解。这通过证明这样的域的分支在值t_j=π/(2√σ_j)处从“平凡”域Ω_{t_j}分叉出来，其中σ_j(j≥1)是拉普拉斯算子在ω⊂R^{N-1}上的简单Neumann特征值。解可以关于ω反射，以生成圆柱中的非平凡解。
英文摘要：
We study an overdetermined eigenvalue problem for domains $Ω$ contained in the half-cylinder $Σ=ω\times (0, +\infty)$, based on a bounded regular domain $ω\subset \mathbb{R}^{N-1}$. It is easy to see that in any bounded cylinder $Ω_{t}=ω\times (0, t)$, $t > 0$, the eigenvalue problem admits a one-dimensional positive eigenfunction which satisfies the overdetermined boundary conditions. The aim of the paper is to construct other domains $Ω\subset Σ$ for which there exists a positive eigenfunction that is a solution of the overdetermined problem. This is achieved by showing that branches of such domains bifurcate from the ``trivial'' domains $Ω_{t_j}$ at the values $t_{j} = \fracπ{2\sqrt{σ_j}}$ where $σ_j$ ($j\geq 1$) is a simple Neumann eigenvalue of the Laplace operator on $ω\subset \mathbb{R}^{N-1}$. The solutions can be reflected with respect to $ω$ to generate nontrivial solutions in a cylinder.
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