论文标题
罗宾和诺伊曼特征值之间的差异
Differences between Robin and Neumann eigenvalues
论文作者
论文摘要
令$ω\ subset \ mathbb r^2 $为有界的平面域，带有分段平滑边界$ \ partialω$。对于$σ> 0 $，我们考虑Robin边界值问题\ [-ΔF=λf，\ Qquad \ frac {\ partial f} {\ partial n} +σf= 0 \ mbox {on} \ partialω\] 其中$ \ frac {\ partial f} {\ partial n} $是向外指向正常指向$ \ partialω$的导数。 令$ 0<λ^σ_0\leqλ^σ_1\ leq \ dots $为相应的特征值。本文的目的是研究罗宾·尼曼（Robin-Neumann）差距 \ [d_n（σ）：=λ_n^σ-λ_n^0。 \]对于一类宽类平面域，我们表明存在一个限制的平均值，等于$ 2 {\ rm长度}（\partialΩ）/{\ rm afore}（ω）\ cdotσ$，在光滑的情况下，给出了$ d_n（σ）\ leq c c（σ）\ leq c c（ω）n^n^n^= $ and and and and and and and obiber。对于沿阵行的台球，我们表明，沿密度一个子序列，差距会汇合到平均值。我们获得了矩形的进一步特性，该矩形具有均匀的上限和磁盘，并改善了一般的上限。
Let $Ω\subset \mathbb R^2$ be a bounded planar domain, with piecewise smooth boundary $\partial Ω$. For $σ>0$, we consider the Robin boundary value problem \[ -Δf =λf, \qquad \frac{\partial f}{\partial n} + σf = 0 \mbox{ on } \partial Ω\] where $ \frac{\partial f}{\partial n} $ is the derivative in the direction of the outward pointing normal to $\partial Ω$. Let $0<λ^σ_0\leq λ^σ_1\leq \dots $ be the corresponding eigenvalues. The purpose of this paper is to study the Robin-Neumann gaps \[ d_n(σ):=λ_n^σ-λ_n^0 . \] For a wide class of planar domains we show that there is a limiting mean value, equal to $2{\rm length}(\partialΩ)/{\rm area}(Ω)\cdot σ$ and in the smooth case, give an upper bound of $d_n(σ)\leq C(Ω) n^{1/3}σ$ and a uniform lower bound. For ergodic billiards we show that along a density-one subsequence, the gaps converge to the mean value. We obtain further properties for rectangles, where we have a uniform upper bound, and for disks, where we improve the general upper bound.
