On the strict monotonicity of the first eigenvalue of the 𝑝-Laplacian on annuli

Anoop, T. V.; Bobkov, Vladimir; Sasi, Sarath

doi:10.1090/tran/7241

On the strict monotonicity of the first eigenvalue of the $p$-Laplacian on annuli
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by T. V. Anoop, Vladimir Bobkov and Sarath Sasi PDF

Trans. Amer. Math. Soc. 370 (2018), 7181-7199 Request permission

Abstract:

Let $B_1$ be a ball in $\mathbb {R}^N$ centred at the origin and let $B_0$ be a smaller ball compactly contained in $B_1$. For $p\in (1, \infty )$, using the shape derivative method, we show that the first eigenvalue of the $p$-Laplacian in annulus $B_1\setminus \overline {B_0}$ strictly decreases as the inner ball moves towards the boundary of the outer ball. The analogous results for the limit cases as $p \to 1$ and $p \to \infty$ are also discussed. Using our main result, further we prove the nonradiality of the eigenfunctions associated with the points on the first nontrivial curve of the Fučik spectrum of the $p$-Laplacian on bounded radial domains.

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