一类非凸集上的Sharp Poincaré不等式
Sharp Poincaré inequalities in a class of non-convex sets
摘要：
设$γ$是一条光滑的、非闭合的、简单的曲线，其图像关于$y$轴对称，并且设$D$是一个平面域，由$γ$一侧上的点组成，在$γ$的适当距离$δ$内。用$μ_1^{odd}(D)$表示最小非平凡Neumann本征值，其对应的本征函数相对于$y$轴为奇数。如果$γ$满足一些简单的几何条件，则$μ_1^{odd}(D)$可以根据$γ$的长度、它的曲率和$δ$从下面精确地估计。此外，我们给出了明确的条件，$δ$，确保$μ_1^{odd}(D)=μ_1(D)$。最后，我们可以将我们在$μ_1^{odd}(D)$上的界推广到某一类三维域上。在二维和三维的设置，我们的域一般是非凸的。
英文摘要：
Let $γ$ be a smooth, non-closed, simple curve whose image is symmetric with respect to the $y$-axis, and let $D$ be a planar domain consisting of the points on one side of $γ$, within a suitable distance $δ$ of $γ$. Denote by $μ_1^{odd}(D)$ the smallest nontrivial Neumann eigenvalue having a corresponding eigenfunction that is odd with respect to the $y$-axis. If $γ$ satisfies some simple geometric conditions, then $μ_1^{odd}(D)$ can be sharply estimated from below in terms of the length of $γ$, its curvature, and $δ$. Moreover, we give explicit conditions on $δ$ that ensure $μ_1^{odd}(D)=μ_1(D)$. Finally, we can extend our bound on $μ_1^{odd}(D)$ to a certain class of three-dimensional domains. In both the two- and three-dimensional settings, our domains are generically non-convex.
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