arXiv subjects

K. Ashok Kumar

Publications and source records attributed to K. Ashok Kumar.

3 recordsLinked to original sources

arXiv2022

Domain variations of the first eigenvalue via a strict Faber-Krahn type inequality

For $d\geq 2$ and $\frac{2d+2}{d+2} < p < \infty $, we prove a strict Faber-Krahn type inequality for the first eigenvalue $λ_1(Ω)$ of the $p$-Laplace operator on a bounded Lipschitz domain $Ω\subset \mathbb{R}^d$ (with mixed boundary conditions) under the polarizations. We apply this inequality to the obstacle problems on the domains of the form $Ω\setminus \mathscr{O}$, where $\mathscr{O}\subset \subset Ω$ is an obstacle. Under some geometric assumptions on $Ω$ and $\mathscr{O}$, we prove the strict monotonicity of $λ_1 (Ω\setminus \mathscr{O})$ with respect to certain translations and rotations of $\mathscr{O}$ in $Ω$.

math.AP↗

arXiv2020

A shape variation result via the geometry of eigenfunctions

We discuss some of the geometric properties, such as the foliated Schwarz symmetry, the monotonicity along the axial and the affine-radial directions, of the first eigenfunctions of the Zaremba problem for the Laplace operator on annular domains. These fine geometric properties, together with the shape calculus, help us to prove that the first eigenvalue is strictly decreasing as the inner ball moves towards the boundary of the outer ball.

math.AP↗

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↗