论文标题
在多边形的Laplacian的第一个特征值上
On the first eigenvalue of the Laplacian for polygons
论文作者
论文摘要
1947年，Pólya证明，如果$ n = 3,4 $，常规Polygon $ p_n $最小化了带有给定面积$α> 0 $的n gon的主要频率，并建议当$ n \ ge 5 $时相同。 $ 1951年，$Pólya＆Szegö讨论了“数学物理学等等的不平等现象”一书中反例的可能性。本文构造了明确的$（2n-4）$ - 尺寸多边形流形$ \ nathcal {m}（m}（n，α）$，并证明存在可计算的$ n \ ge 5 $，这样可以通过$ \ n $ expllic and $ \ mathcal and $ n $ a; \ Mathcal {a} _ {n}（α）\ subset \ mathcal {m}（n，α）$，以至于$ p_n $在$ n $ gons中具有$ n $ gon的最小主频率，$ \ nathcal {a} _} _ {n}（n}（α）$。当$ n \ ge 3 $时，在\ Mathcal {m} in \ Mathcal {m}（n，α）$的主要频率的主要频率中证明了一个公式。而且，事实证明，一组等边的多边形被证明是$（n-3）$ - $（2N-4）$的尺寸submanifold-尺寸歧管$ \ MATHCAL {m}（m}（n，α）$附近$ p_n $。如果$ n = 3 $，则该公式完全解决了2006年的抗脉和频率的猜想，以及“数学物理学的等等不平等现象”中提到的另一个问题。此外，给出了三角形的尖锐多边形faber-krahn稳定性问题的解决方案，并具有明确的常数。这些技术涉及部分对称，张量计算，循环矩阵的光谱理论和$ w^{2，p}/bmo $估计。最后，在电子气泡的上下文中给出了应用。
In 1947, Pólya proved that if $n=3,4$ the regular polygon $P_n$ minimizes the principal frequency of an n-gon with given area $α>0$ and suggested that the same holds when $n \ge 5$. In $1951,$ Pólya & Szegö discussed the possibility of counterexamples in the book "Isoperimetric Inequalities In Mathematical Physics." This paper constructs explicit $(2n-4)$--dimensional polygonal manifolds $\mathcal{M}(n, α)$ and proves the existence of a computable $N \ge 5$ such that for all $n \ge N$, the admissible $n$-gons are given via $\mathcal{M}(n, α)$ and there exists an explicit set $ \mathcal{A}_{n}(α) \subset \mathcal{M}(n,α)$ such that $P_n$ has the smallest principal frequency among $n$-gons in $\mathcal{A}_{n}(α)$. Inter-alia when $n \ge 3$, a formula is proved for the principal frequency of a convex $P \in \mathcal{M}(n,α)$ in terms of an equilateral $n$-gon with the same area; and, the set of equilateral polygons is proved to be an $(n-3)$--dimensional submanifold of the $(2n-4)$--dimensional manifold $\mathcal{M}(n,α)$ near $P_n$. If $n=3$, the formula completely addresses a 2006 conjecture of Antunes and Freitas and another problem mentioned in "Isoperimetric Inequalities In Mathematical Physics." Moreover, a solution to the sharp polygonal Faber-Krahn stability problem for triangles is given and with an explicit constant. The techniques involve a partial symmetrization, tensor calculus, the spectral theory of circulant matrices, and $W^{2,p}/BMO$ estimates. Last, an application is given in the context of electron bubbles.
