论文标题
非线性椭圆算子的第一个罗宾特征值的尖锐估计值
Sharp estimates for the first Robin eigenvalue of nonlinear elliptic operators
论文作者
论文摘要
本文的目的是获得各向异性$ p $ -laplace运算符的第一个罗宾特征值的最佳估计，即：\ begin {qore*}λ_1（β，β，ω）= \ min_ = \ min_ {ψ\ in w^{1，p}（p}（p}（p}（p）， \ frac {\ displayStyle \int_Ωf（\nablaψ）^p dx +β\ displayStyle \ int \ int _ {\partialΩ} |ψ|^|^pf（ν_Ω）d \ MATHCAL H^{n-1}}}} $ p \ in] 1，+\ infty [$，$ω$是一个有界的，平均的凸域中的$ \ Mathbb r^{n} $，$ν_Ω$是其euclidean euclidean Out norryal，$β$是一个真实的数字，$ f $是$ f $的足够平稳的规范。我们发现的估计值是根据一维非线性问题的第一个特征值，它取决于$β$以及与$ω$相关的几何数量。更确切地说，在情况下，我们证明了$λ_{1} $的下限$β> 0 $，在情况下是$β<0 $的上限。结果，我们以$β> 0 $的价格证明，在各向异性inradius的$λ_{1}（β，ω）$的下限为$ω$，并且，$β<0 $，$λ_{1}（β，β，ω）$ $β$ $β$的上限。
The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic $p$-Laplace operator, namely: \begin{equation*} λ_1(β,Ω)=\min_{ψ\in W^{1,p}(Ω)\setminus\{0\} } \frac{\displaystyle\int_ΩF(\nabla ψ)^p dx +β\displaystyle\int_{\partialΩ}|ψ|^pF(ν_Ω) d\mathcal H^{N-1} }{\displaystyle\int_Ω|ψ|^p dx}, \end{equation*} where $p\in]1,+\infty[$, $Ω$ is a bounded, mean convex domain in $\mathbb R^{N}$, $ν_Ω$ is its Euclidean outward normal, $β$ is a real number, and $F$ is a sufficiently smooth norm on $\mathbb R^{N}$. The estimates we found are in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on $β$ and on geometrical quantities associated to $Ω$. More precisely, we prove a lower bound of $λ_{1}$ in the case $β>0$, and a upper bound in the case $β<0$. As a consequence, we prove, for $β>0$, a lower bound for $λ_{1}(β,Ω)$ in terms of the anisotropic inradius of $Ω$ and, for $β<0$, an upper bound of $λ_{1}(β,Ω)$ in terms of $β$.
