arXiv subjects
Publications and source records attributed to Sheela Verma.
In this article, we prove a variant of domain monotonicity property for the Steklov eigenvalues of the Laplacian on strictly star-shaped bounded domains contained in $\mathbb{R}^n$. As an application of this result, we obtain an upper and a lower bound for Steklov eigenvalues on convex domains.
In this article, we study the mixed Steklov--Neumann eigenvalue problem on doubly connected domains. First, we show that among all doubly connected domains in $\mathbb{R}^n$ of the form $B_{R_2}\setminus \overline{B_{R_1}}$, where $B_{R_1}$ and $B_{R_2}$ are open balls of fixed radii satisfying $\overline{B_{R_1}} \subset B_{R_2}$, the first non-zero Steklov--Neumann eigenvalue attains its maximal value when the balls are concentric. Next, we establish bounds for the first non-zero Steklov--Neumann eigenvalue on a doubly connected star-shaped domain contained in a hypersurface equipped with a revolution-type metric. We also derive the asymptotic behavior of the first non-zero Steklov--Neumann eigenvalue on a bounded domain with a spherical hole in $\mathbb{R}^n$ as the radius of the hole approaches zero. Finally, we study the number of nodal domains of the eigenfunction corresponding to the first non zero Steklov--Neumann eigenvalue on a bounded domain in $\mathbb{R}^n$ having a spherical hole.
Let $\Omega$ be a bounded open set in the Poincar\'e hyperbolic disk, $\mathbb{D}$. In this article, we consider the hyperbolic logarithmic potential operator $\mathcal{L}_h : L^2(\Omega) \to L^2(\Omega)$, defined by \begin{equation*} \mathcal{L}_h u(z)=\frac{1}{2}\int_\Omega \log\frac{1}{[z,w]}\,u(w)\, {\,\rm d}(w), \end{equation*} and the associated eigenvalue problem on $\Omega$ \begin{equation} \mathcal{L}_h u=\tau u. \end{equation} We first extend the notion of polarization with respect to hyperplanes in the Poincar\'e disk and prove the associated properties. Then we establish a reverse Faber-Krahn inequality for the largest eigenvalue, $\tau_{h}$ of $\mathcal{L}_h$, under polarization. Further, we provide a representation formula for the eigenfunctions of $\mathcal{L}_h$. In addition, we show that the operator $\mathcal{L}_h$ is a positive operator on $L^2(\Omega)$.
In this article, we study sharp bounds for the Neumann eigenvalues of the Laplace operator on graphs. We first establish both lower and upper bounds for the second Neumann eigenvalue on simple graphs, and then derive monotonicity properties of Neumann eigenvalues on trees. In particular, we show that adding a vertex to a tree reduces the corresponding Neumann eigenvalues. We wish to emphasize that monotonicity results for Neumann eigenvalues on trees already exist in the literature, but our proof follows a fundamentally different approach. As a consequence of this monotonicity result, we provide an upper bound for the second Neumann eigenvalue and a lower bound for the largest Neumann eigenvalue on trees. Finally, we prove that under a diameter constraint on trees, the largest Neumann eigenvalue cannot be bounded from above.
In this article, we study Steklov eigenvalues and mixed Steklov Neumann eigenvalues on a smooth bounded domain in $\mathbb{R}^{n}$, $n \geq 2$, having a spherical hole. We focus on two main results related to Steklov eigenvalues. First, we obtain explicit expression for the second nonzero Steklov eigenvalue on concentric annular domain. Secondly, we derive a sharp upper bound of the first $n$ nonzero Steklov eigenvalues on a domain $\Omega \subset \mathbb{R}^{n}$ having symmetry of order $4$ and a ball removed from its center. This bound is given in terms of the corresponding Steklov eigenvalues on a concentric annular domain of the same volume as $\Omega$. Next, we consider the mixed Steklov Neumann eigenvalue problem on $4^{\text{th}}$ order symmetric domains in $\mathbb{R}^{n}$ having a spherical hole and obtain upper bound of the first $n$ nonzero eigenvalues. We also provide some examples to illustrate that symmetry assumption in our results is crucial. Finally, We make some numerical observations about these eigenvalues using FreeFEM++ and state them as conjectures.
We consider mixed Steklov-Dirichlet eigenvalue problem on smooth bounded domains in Riemannian manifolds. Under certain symmetry assumptions on multiconnected domains in $\mathbb{R}^{n}$ with a spherical hole, we obtain isoperimetric inequalities for $k$-th Steklov-Dirichlet eigenvalues for $2 \leq k \leq n+1$. We extend Theorem 3.1 of \cite{gavitone2023isoperimetric} from Euclidean domains to domains in space forms, that is, we obtain sharp lower and upper bounds of the first Steklov-Dirichlet eigenvalue on bounded star-shaped domains in the unit $n$-sphere and in the hyperbolic space.
In this note, an isoperimetric inequality for the harmonic mean of lower order Steklov eigenvalues is proved on bounded domains in noncompact rank-$1$ symmetric spaces. This work extends result of \cite{BR.01} and \cite{V.21} proved for bounded domains in Euclidean space and Hyperbolic space, respectively.
In this paper, we study the shape optimization problem for the first eigenvalue of the $p$-Laplace operator with the mixed Neumann-Dirichlet boundary conditions on multiply-connected domains in hyperbolic space. Precisely, we establish that among all multiply-connected domains of a given volume and prescribed $(n-1)$-th quermassintegral of the convex Dirichlet boundary (inner boundary), the concentric annular region produces the largest first eigenvalue. We also derive Nagy's type inequality for outer parallel sets of a convex domain in the hyperbolic space.
We consider the Neumann eigenvalue problem for Laplacian on a bounded multi-connected domain contained in simply connected space forms. Under certain symmetry assumptions on the domain, we prove Szeg\"{o}-Weinberger type inequalities for the first $n$ positive Neumann eigenvalues.
In this note, we find a sharp upper bound for the Steklov spectrum on a submanifold of revolution in Euclidean space with one boundary component.
In this article, we prove an isoperimetric inequality for the harmonic mean of the first $(n-1)$ nonzero Steklov eigenvalues on bounded domains in $n$-dimensional Hyperbolic space. Our approach to prove this result also gives a similar inequality for the first $n$ nonzero Steklov eigenvalues on bounded domains in $n$-dimensional Euclidean space.
Let $\mathbb{M}$ denote a complete, simply connected Riemannian manifold with sectional curvature $K_{\mathbb{M}} \leq k$ and Ricci curvature $\text{Ric}_{\mathbb{M}} \geq (n-1)K$, where $k,K \in \mathbb{R}$. Then for a bounded domain $\Omega \subset\mathbb{M}$ with smooth boundary, we prove that the first nonzero Neumann eigenvalue $\mu_{1}(\Omega) \leq \mathcal{C} \mu_{1}(B_{k}(R))$. Here $B_{k}(R)$ is a geodesic ball of radius $R > 0$ in the simply connected space form $\mathbb{M}_{k}$ such that vol$(\Omega)$ = vol$(B_{k}(R))$, and $\mathcal{C}$ is a constant which depends on the volume, diameter of $\Omega$ and the dimension of $\mathbb{M}$.
This work is an extension of a result given by Kuttler and Sigillito (SIAM Rev $10$:$368-370$, $1968$) on a star-shaped bounded domain in $\mathbb{R}^2$. Let $\Omega$ be a star-shaped bounded domain in a hypersurface of revolution, having smooth boundary. In this article, we obtain a sharp lower bound for all Steklov eigenvalues on $\Omega$ in terms of the Steklov eigenvalues of the largest geodesic ball contained in $\Omega$ with the same center as $\Omega$. We also obtain similar bounds for all Steklov eigenvalues on star-shaped bounded domain in paraboloid, $P = \left\lbrace (x, y, z) \in \mathbb{R}^{3} : z = x^2 + y^2\right\rbrace$.
Let $M$ be a closed hypersurface in $\mathbb{R}^{n}$ and $\Omega$ be a bounded domain such that $M= \partial\Omega$. In this article, we obtain an upper bound for the first non-zero eigenvalue of the following problems. \begin{itemize} \item Closed eigenvalue problem: \begin{align*} %\label{eqn:closedep} \Delta_p u = \lambda_{p} \ |u|^{p-2} \ u \qquad \mbox{ on } \quad {M}. \end{align*} \item Steklov eigenvalue problem: \begin{align*} \begin{array}{rcll} \Delta_{p}u &=& 0 & \mbox{ in } \Omega ,\\ |\nabla u|^{p-2} \frac{\partial u}{\partial \nu} &=& \mu_{p} \ |u|^{p-2} \ u &\mbox{ on } M . \end{array} \end{align*} \end{itemize}
We consider two eigenvalue problems for Laplacian on some specific doubly connected domain. In particular, we study the following two eigenvalue problems. Let $B_1$ be an open ball in $\mathbb{R}^n$ and $B_0$ be a ball contained in $B_1$. Let $\nu$ be the outward unit normal on $\partial B_1$. Then the first eigenvalue of the problem \begin{align*} \begin{array}{rcll} \Delta u &=& 0 \, &\mbox{ in } \, B_1 \setminus \bar{B}_0 , \\ u &=& 0 \, &\mbox{ on } \, {\partial B_0}, \\ \frac{\partial u}{\partial \nu} &=& \tau \, u \, &\mbox{ on } \, {\partial B_1}, \end{array} \end{align*} attains maximum if and only if $B_0$ and $B_1$ are concentric. Let $D$ be a domain in a non-compact rank-$1$ symmetric space $(\mathbb{M}, ds^2)$, geodesically symmetric with respect to the point $ p\in \mathbb{M}$. Let $B_0$ be a ball in $\mathbb{M}$ centered at $p$ such that $\bar{{B}_0}\subset D$ and $\nu$ be the outward unit normal on ${\partial (D \setminus \bar{B}_0)}$. Then the first non-zero eigenvalue of \begin{align*} \begin{array}{rcll} \Delta u &=& \mu \ u \, &\mbox{ in } \, D \setminus \bar{B}_0, \\ \frac{\partial u}{\partial \nu} &=& 0 \, &\mbox{ on } \, {\partial (D \setminus \bar{B}_0)}, \end{array} \end{align*} attains maximum if and only if $D$ is a geodesic ball centered at $p$.
Let $\Omega$ be a star-shaped bounded domain in $(\mathbb{S}^{n}, ds^{2})$ with smooth boundary. In this article, we give a sharp lower bound for the first non-zero eigenvalue of the Steklov eigenvalue problem in $\Omega.$ This result is the generalization of a result given by Kuttler and Sigillito for a star-shaped bounded domain in $\mathbb{R}^2.$ Further we also obtain a two sided bound for the first non-zero eigenvalue of the Steklov problem on the ball in $\mathbb{R}^n$ with rotationally invariant metric and with bounded radial curvature.