An eigenvalue estimate for a Robin 𝑝-Laplacian in 𝐶¹ domains,Proceedings of the American Mathematical Society

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An eigenvalue estimate for a Robin 𝑝-Laplacian in 𝐶¹ domains
Proceedings of the American Mathematical Society ( IF 0.8 ) Pub Date : 2020-06-30 , DOI: 10.1090/proc/15116
Konstantin Pankrashkin

Abstract:Let $\Omega \subset \mathbb{R}^n$ be a bounded

domain and

. For $\alpha >0$ , define the quantity

$\displaystyle \Lambda (\alpha )=\inf _{u\in W^{1,p}(\Omega ),\, u\not \equiv 0}... ...\vert^p \,\mathrm {d} s\Big )\Big / \int _\Omega \vert u\vert^p\,\mathrm {d} x$

with $\mathrm {d} s$ being the hypersurface measure, which is the lowest eigenvalue of the

-Laplacian in $\Omega$ with a non-linear $\alpha$ -dependent Robin boundary condition. We show the asymptotics $\Lambda (\alpha )=(1-p)\alpha ^{p/(p-1)}+o(\alpha ^{p/(p-1)})$ as $\alpha$ tends to $+\infty$ . The result was only known for the linear case

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 $C^{1,\lambda }$ domains.

中文翻译：

𝐶领域中Robin𝑝-Laplacian的特征值估计

摘要：让一个有界域和。对于，定义数量 $\ Omega \ subset \ mathbb {R} ^ n$

$\ alpha> 0$

$\ displaystyle \ Lambda（\ alpha）= \ inf _ {u \ in W ^ {1，p}（\ Omega），\，u \ not \ equiv 0} ... ... \ vert ^ p \， \ mathrm {d} s \ Big）\ Big / \ int _ \ Omega \ vert u \ vert ^ p \，\ mathrm {d} x$

与作为超曲面量度，这是最低的本征值-Laplacian在与非线性依赖性罗宾边界条件。我们展示了渐进性的趋向。仅在线性情况下或在更强的平滑度假设下才知道结果。我们的证明要短得多，并且基于完全不同的基本参数，并且可以改进域的余数估计。 $\ mathrm {d} s$

$\ Omega$ $\ alpha$ $\ Lambda（\ alpha）=（1-p）\ alpha ^ {p /（p-1）} + o（\ alpha ^ {p /（p-1）}）$ $\ alpha$ $+ \ infy$

$C ^ {1，\ lambda}$

更新日期：2020-09-01

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