arXiv subjects

T. V. Anoop

Publications and source records attributed to T. V. Anoop.

21 records · Page 2Linked to original sources

arXiv2019

On the reverse Faber-Krahn inequalities

Payne-Weinberger showed that \textit{`among the class of membranes with given area $A$, free along the interior boundaries and fixed along the outer boundary of given length $L_0$, the annulus $Ω^\#$ has the highest fundamental frequency,'} where $Ω^\#$ is a concentric annulus with the same area as $Ω$ and the same outer boundary length as $L_0$. We extend this result for the higher dimensional domains and $p$-Laplacian with $p\in (1,\infty),$ under the additional assumption that the outer boundary is a sphere. As an application, we prove that the nodal set of the second eigenfunctions of $p$-Laplacian (with mixed boundary conditions) on a ball and a concentric annulus cannot be a concentric sphere.

math.AP↗

arXiv2016

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

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.

math.AP↗

arXiv2015

On the structure of the second eigenfunctions of the p-Laplacian on a ball

In this paper, we prove that the second eigenfunctions of the $p$-Laplacian, $p>1$, are not radial on the unit ball in $\mathbb{R}^N,$ for any $N\ge 2.$ Our proof relies on the variational characterization of the second eigenvalue and a variant of the deformation lemma. We also construct an infinite sequence of eigenpairs $\{τ_n,Ψ_n\}$ such that $Ψ_n$ is nonradial and has exactly $2n$ nodal domains. A few related open problems are also stated.

math.AP↗