On the strict monotonicity of the first eigenvalue of the $p$Laplacian on annuli
Abstract
Let $B_1$ be a ball in $\mathbb{R}^N$ centred at the origin and $B_0$ 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2016
 arXiv:
 arXiv:1611.03532
 Bibcode:
 2016arXiv161103532A
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Optimization and Control;
 35J92;
 35P30;
 35B06;
 49R05
 EPrint:
 19 pages