论文标题
尖锐无限锥上罗宾特征值的渐近学
Asymptotics of Robin eigenvalues on sharp infinite cones
论文作者
论文摘要
令$ω\ subset \ mathbb {r}^n $为Lipschitz边界的有限域。对于$ \ varepsilon> 0 $和$ n \ in \ mathbb {n} $考虑无限锥$ω_ {\ varepsilon}：= \ big \ {（x_1，x'）\ in（in（x_1，x'） x_1Ω\ big \} \ subset \ mathbb {r}^{n+1} $和操作员$ q _ {\ varepsilon}^α$用作laplacian $ \ u \ u \ u \ mapsto-Δu$ on $ { $ \partialΩ_\ varepsilon $，其中$ \partial_ν$是外向的普通导数，$α> 0 $。我们查看$ q_ \ varepsilon^α$对参数$ \ varepsilon $的特征值的依赖性：以前仅以$ n = 1 $解决了此问题（在这种情况下，唯一可接受的$ω$是有限间隔）。在目前的工作中，我们考虑任意尺寸$ n \ ge2 $并任意形状的“跨式” $ω$，然后将光谱渐近性查看为$ \ varepsilon $变小，即圆锥体变得“尖锐”并倒入半线。事实证明，单个特征值的渐近学的主要术语由单个几何数量$n_Ω：= \ dfrac {\ mathrm {vol} _ {n-1} _ {n-1} \partialΩ} {\ mathrm {vol} _n _nω} $。更准确地说，对于任何固定的$ j \ in \ mathbb {n} $和$α> 0 $ j $ j $ th eigenvalue $ e_j（q^α__\ varepsilon）$ q^α__\ varepsilon $ of $ q^α_\ VAREPSILON $都有足够的小$ \ varepsilon uss and Explast $ e_j（q^α_\ varepsilon）= - \ dfrac {n_Ω^2 \，α^2} {（（2J+n-2）^2 \，\ \ varepsilon^2}+o \ lest（\ dfrac {\ dfrac {\ dfrac {1} {\ varepsilon} {\ varepsilon} {\ right）该论文还涵盖了无限锥上Sobolev空间的某些方面，这可能具有独立感兴趣。
Let $ω\subset\mathbb{R}^n$ be a bounded domain with Lipschitz boundary. For $\varepsilon>0$ and $n\in\mathbb{N}$ consider the infinite cone $Ω_{\varepsilon}:=\big\{(x_1,x')\in (0,\infty)\times\mathbb{R}^n: x'\in\varepsilon x_1ω\big\}\subset\mathbb{R}^{n+1}$ and the operator $Q_{\varepsilon}^α$ acting as the Laplacian $u\mapsto-Δu$ on $Ω_{\varepsilon}$ with the Robin boundary condition $\partial_νu=αu$ at $\partialΩ_\varepsilon$, where $\partial_ν$ is the outward normal derivative and $α>0$. We look at the dependence of the eigenvalues of $Q_\varepsilon^α$ on the parameter $\varepsilon$: this problem was previously addressed for $n=1$ only (in that case, the only admissible $ω$ are finite intervals). In the present work we consider arbitrary dimensions $n\ge2$ and arbitrarily shaped "cross-sections" $ω$ and look at the spectral asymptotics as $\varepsilon$ becomes small, i.e. as the cone becomes "sharp" and collapses to a half-line. It turns out that the main term of the asymptotics of individual eigenvalues is determined by the single geometric quantity $N_ω:=\dfrac{\mathrm{Vol}_{n-1} \partialω}{\mathrm{Vol}_n ω}$. More precisely, for any fixed $j\in \mathbb{N}$ and $α>0$ the $j$th eigenvalue $E_j(Q^α_\varepsilon)$ of $Q^α_\varepsilon$ exists for all sufficiently small $\varepsilon>0$ and satisfies $E_j(Q^α_\varepsilon)=-\dfrac{N_ω^2\,α^2}{(2j+n-2)^2\,\varepsilon^2}+O\left(\dfrac{1}{\varepsilon}\right)$ as $\varepsilon\to 0^+$. The paper also covers some aspects of Sobolev spaces on infinite cones, which can be of independent interest.
