中科院数学与系统科学研究院
数学研究所
学术报告
非线性分析研讨班
报告人：Emanuel Indrei （Purdue University）
题 目：On the first eigenvalue of the Laplacian for polygons
时 间：2022.12.16（星期五）9:00-11:00
地 点：ZOOM ID: 4120194771 Password: mcm1234
摘 要：In 1947, P\'olya proved that if $n=3,4$ the regular polygon $P_n$ minimizes the principal frequency of an n-gon with given area $\alpha>0$ and suggested that the same holds when $n \ge 5$. In $1951,$ P\'olya \& Szeg\"o discussed the possibility of counterexamples in the book ``Isoperimetric Inequalities In Mathematical Physics." My recent work constructs a polygonal space $\mathcal{A}_{n}(\alpha)$ when $n$ is large such that $P_n$ has the smallest principal frequency among $n$-gons in $\mathcal{A}_{n}(\alpha)$. Inter-alia when $n \ge 3$, I obtained a formula for the principal frequency of a convex $P$ in terms of an equilateral $n$-gon with the same area. If $n=3$, the formula completely addresses a conjecture of Antunes and Freitas and another problem mentioned in ``Isoperimetric Inequalities In Mathematical Physics." Moreover, I in addition obtained a solution to the sharp polygonal Faber-Krahn stability problem for triangles 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, assuming $n$ is large, I obtained the scaling of the regular polygon thanks to a 4th order equation in the context of the problem of minimizing an energy for electron bubbles.
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