论文标题
罗宾$ p $ -laplacian in $ c^1 $域中的特征值估计
An eigenvalue estimate for a Robin $p$-Laplacian in $C^1$ domains
论文作者
论文摘要
令$ω\ subset \ mathbb {r}^n $为有限的$ c^1 $域，$ p> 1 $。对于$α> 0 $，定义数量\ [λ（α）= \ inf_ {u \ in w^{1，p}（p}（ω），\，\，u \ equiv 0} \ big（\int_Ω \ \，\ Mathrm {d} s \ big）\ big/ \ big/ \int_Ω| u |^p \，\ mathrm {d} x \]，$ \ mathrm {d} s $是高超纯度措施，这是$ p $ -laplacian in $ p $ -laplacian in a $ obin $ obin $ a $ - $ - $ - $ - $ - $ - $ - $ - 我们将渐近性$λ（α）=（1-p）α^{p/（p-1）}+o（α^{p/（p/（p-1）}）$作为$α$倾向于$+\ infty $。结果仅因线性案例$ p = 2 $或更强的平滑度假设而闻名。我们的证明要短得多，并且基于完全不同和基本的参数，并且可以对$ c^{1，λ} $域进行改进的剩余估计。
Let $Ω\subset \mathbb{R}^n$ be a bounded $C^1$ domain and $p>1$. For $α>0$, define the quantity \[ Λ(α)=\inf_{u\in W^{1,p}(Ω),\, u\not\equiv 0} \Big(\int_Ω|\nabla u|^p\,\mathrm{d}x - α\int_{\partialΩ} |u|^p \,\mathrm{d} s\Big)\Big/ \int_Ω|u|^p\,\mathrm{d} x \] with $\mathrm{d} s$ being the hypersurface measure, which is the lowest eigenvalue of the $p$-laplacian in $Ω$ with a non-linear $α$-dependent Robin boundary condition. We show the asymptotics $Λ(α) =(1-p)α^{p/(p-1)}+o(α^{p/(p-1)})$ as $α$ tends to $+\infty$. The result was only known for the linear case $p=2$ or under stronger smoothness assumptions. Our proof is much shorter and is based on completely different and elementary arguments, and it allows for an improved remainder estimate for $C^{1,λ}$ domains.
