论文标题
第一个非零steklov特征值的上限
An upper bound for the first nonzero Steklov eigenvalue
论文作者
论文摘要
令$（m^n，g）$为完整的简单连接的$ n $二维riemannian歧管，具有曲率界限$ \ permatatorname {sect} _g \ leq \ leqκ$ for $κ\ leq 0 $ and $ \ leq 0 $ and $ \ operatorname {ric} _g \ geq（n-1）我们证明，对于任何有限的域$ω\ subset m^n $带直径$ d $和Lipschitz边界，如果$ω^*$是一个简单连接的空间形式的地理球，则具有恒定的截面曲率$κ$封装的$ω$，然后是$ c c cccent $ qucccent $ qucccent $ qum和$σ_1$ qum和$σ_1（pher） $σ_1（ω^*）$分别为$ω$和$ω^*$的第一个非零steklov特征值，$ c = c = c（n，κ，k，d）$是一个明确的常数。当$κ= k $时，我们有$ C = 1 $并恢复Brock-Weinstock的不平等现象，并断言地理球在同一体积的域中，在欧几里得空间和多重增多空间中独特地最大化了第一个非零的steklov特征值。
Let $(M^n,g)$ be a complete simply connected $n$-dimensional Riemannian manifold with curvature bounds $\operatorname{Sect}_g\leq κ$ for $κ\leq 0$ and $\operatorname{Ric}_g\geq(n-1)Kg$ for $K\leq 0$. We prove that for any bounded domain $Ω\subset M^n$ with diameter $d$ and Lipschitz boundary, if $Ω^*$ is a geodesic ball in the simply connected space form with constant sectional curvature $κ$ enclosing the same volume as $Ω$, then $σ_1(Ω) \leq C σ_1(Ω^*)$, where $σ_1(Ω)$ and $ σ_1(Ω^*)$ denote the first nonzero Steklov eigenvalues of $Ω$ and $Ω^*$ respectively, and $C=C(n,κ, K, d)$ is an explicit constant. When $κ=K$, we have $C=1$ and recover the Brock-Weinstock inequality, asserting that geodesic balls uniquely maximize the first nonzero Steklov eigenvalue among domains of the same volume, in Euclidean space and the hyperbolic space.
