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T. V. Anoop

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

At least 19 recordsLinked to original sources
arXiv2026

On the Hersch-Weinberger inequality in higher dimensions

We investigate a reverse Faber-Krahn type inequality for the Robin Laplacian in a bounded smooth domain $\Omega \subset \mathbb{R}^N$ whose boundary has two connected components. We prove that a concentric spherical shell maximizes the first eigenvalue over a class of such domains under perimeter and volume constraints, and under an additional convexity assumption when $N \geq 3$. This result generalizes to a wider class, and extends to higher dimensions, the inequality of Hersch [20], whose approach was substantially based on a construction of the so-called effectless cut by Weinberger [35], so that we call it the Hersch-Weinberger inequality. Our method is based on the analysis of the gradient flow of the first eigenfunction and several approximation procedures, without relying on the effectless cut itself. The effectless cut being a complicated object related to the attractor of the gradient flow, we describe its most fundamental topological properties. In particular, we show that it does not necessarily have to be a hypersurface.

math.AP↗
arXiv2026

On the weighted logarithmic potential operator

For a bounded open set $\Omega \subset \mathbb{R}^N$ with $N\geq 2$, and for positive continuous functions $w,g$ on $\overline{\Omega}$, we consider the weighted eigenvalue problem \begin{equation*} \mathcal{L}_{w} u =\tau gu, \end{equation*} where $\mathcal{L}_{w}$ is the weighted logarithmic potential operator on $L^2(\Omega)$ as defined below: \begin{equation*} \mathcal{L}_{w} u(x)=\int_\Omega \log\left(\frac{w(x)w(y)}{|x-y|}\right)u(y)dy. \end{equation*} We study the monotonicity and continuity of the largest positive eigenvalue $\tau_{w,g}^+(\Omega)$ with respect to $\Omega$, $w$, and $g$. We also establish that $\tau_{w,g}^+(\Omega)$ satisfies a reverse Faber Krahn inequality under polarization. We provide a sufficient condition for the existence of a negative eigenvalue in terms of the weighted transfinite diameter of $\Omega$, under the assumption that $\log w$ is superharmonic. For $\Omega\subset \mathbb{R}^2$, if $\Delta\log w $ is a constant $C$, we show that 0 can be an eigenvalue of $\mathcal{L}_{w}$ only when $C=\frac{2\pi}{|\Omega|}$. For such domains, if $\log w$ is a harmonic function on $\Omega$, we provide a representation formula for the eigenfunctions. Using this representation, we establish variants of the maximum principles that give some insight into the geometry of these eigenfunctions.

math.AP↗
arXiv2025

Reverse Faber-Krahn inequality for planar doubly connected domains

We prove that among all doubly connected and elastically supported planar membranes $\Omega$ with prescribed values of the area $|\Omega|$ and the lengths of the inner and outer boundaries $|\partial \Omega_{\rm{in}}|_1$, $|\partial \Omega_{\rm{out}}|_1$ satisfying $|\partial \Omega_{\rm{out}}|_1^2 - |\partial \Omega_{\rm{in}}|_1^2 = 4\pi |\Omega|$, the concentric annular membrane has the maximal fundamental frequency. The elastic constants $h_{\rm{in}}$, $h_{\rm{out}}$ on $\partial \Omega_{\rm{in}}$, $\partial \Omega_{\rm{out}}$, respectively, are assumed to satisfy $h_{\rm{in}} \cdot h_{\rm{out}} \geq 0$ and can admit negative values and $+\infty$, the latter being understood as a fixation of the membrane on the corresponding part of the boundary. Our study extends and unifies several existing results in the literature. The case $h_{\rm{in}} \cdot h_{\rm{out}} = 0$ is proved using the method of interior parallels \`a la Payne & Weinberger, and it requires less restrictive assumptions on $\Omega$. For the case $h_{\rm{in}} \cdot h_{\rm{out}} > 0$, we develop the construction of the so-called ``effectless cut'' of $\Omega$ described in terms of the gradient flow of the first eigenfunction. This concept was originally introduced by Weinberger and used by Hersch in the fixed boundary case, whose arguments we also revise.

math.AP↗
arXiv2025

Reverse Faber-Krahn inequalities for the Logarithmic potential operator

For a bounded open set $\Omega \subset \mathbb{R}^2,$ we consider the largest eigenvalue $\tau_1(\Omega)$ of the Logarithmic potential operator $\mathcal{L}$. If $diam(\Omega)\le 1$, we prove reverse Faber-Krahn type inequalities for $\tau_1(\Omega)$ under polarization and Schwarz symmetrization. Further, we establish the monotonicity of $\tau_1(\Omega\setminus\mathcal{O})$ with respect to certain translations and rotations of the obstacle $\mathcal{O}$ within $\Omega$. The analogous results are also stated for the largest eigenvalue of the Riesz potential operator. Furthermore, we investigate properties of the smallest eigenvalue $\tilde{\tau}_1(\Omega)$ for a domain whose transfinite diameter is greater than 1. Finally, we characterize the eigenvalues of $\mathcal{L}$ on $B_R$, including the $\tilde{\tau}_1(B_R)$ when $R>1$.

math.AP↗
arXiv2024

Neumann domains of planar analytic eigenfunctions

Along with the partition of a planar bounded domain $\Omega$ by the nodal set of a fixed eigenfunction of the Laplace operator in $\Omega$, one can consider another natural partition of $\Omega$ by, roughly speaking, gradient flow lines of a special type (separatrices) of this eigenfunction. Elements of such partition are called Neumann domains and their boundaries are Neumann lines. When the eigenfunction is a Morse function, this partition corresponds to the Morse--Smale complex and its fundamental properties have been systematically investigated by Band & Fajman (2016). Although, in the case of general position, eigenfunctions are always of the Morse type, particular eigenfunctions can possess degenerate critical points. In the present work, we propose a way to characterize Neumann domains and lines of an arbitrary eigenfunction. Instead of requiring the nondegeneracy of critical points of the eigenfunction, its real analyticity is principally used. The analyticity allows for the presence of degenerate critical points but significantly limits their possible diversity. Even so, the eigenfunction can possess curves of critical points, which have to belong naturally to the Neumann lines set, as well as critical points of a saddle-node type. We overview all possible types of degenerate critical points in the eigenfunction's critical set and provide a numerically based evidence that each of them can be observed for particular eigenfunctions. Alongside with [Band & Fajman, 2016], our approach is inspired by a little-known note of Weinberger that appeared back in 1963, where a part of the Neumann line set, under the name of "effectless cut", was explicitly introduced and studied for the first eigenfunctions in domains with nontrivial topology. In addition, we provide an asymptotic counting of Neumann domains for a disk and rectangles in analogy with the Pleijel constant.

math.AP↗
arXiv2024

On Fractional Orlicz-Hardy Inequalities

We establish the weighted fractional Orlicz-Hardy inequalities for various Orlicz functions. Further, we identify the critical cases for each Orlicz function and prove the weighted fractional Orlicz-Hardy inequalities with logarithmic correction. Moreover, we discuss the analogous results in the local case. In the process, for any Orlicz function $\Phi$ and for any $\Lambda>1$, the following inequality is established $$ \Phi(a+b)\leq \lambda\Phi(a)+\frac{C( \Phi, \Lambda )}{(\lambda-1)^{p_\Phi^+-1}}\Phi(b),\;\;\;\forall\,a,b\in [0,\infty),\,\forall\,\lambda\in (1,\Lambda], $$ where $p_\Phi^+:=\sup\big\{t\varphi(t)/\Phi(t):t>0\big\},$ $\varphi$ is the right derivatives of $\Phi$ and $C( \Phi, \Lambda )$ is a positive constant that depends only on $\Phi$ and $\Lambda.$

math.AP↗
arXiv2023

Reverse Faber-Krahn and Szego-Weinberger type inequalities for annular domains under Robin-Neumann boundary conditions

Let $\tau_k(\Omega)$ be the $k$-th eigenvalue of the Laplace operator in a bounded domain $\Omega$ of the form $\Omega_{\text{out}} \setminus \overline{B_{\alpha}}$ under the Neumann boundary condition on $\partial \Omega_{\text{out}}$ and the Robin boundary condition with parameter $h \in (-\infty,+\infty]$ on the sphere $\partial B_\alpha$ of radius $\alpha>0$ centered at the origin, the limiting case $h=+\infty$ being understood as the Dirichlet boundary condition on $\partial B_\alpha$. In the case $h>0$, it is known that the first eigenvalue $\tau_1(\Omega)$ does not exceed $\tau_1(B_\beta \setminus \overline{B_\alpha})$, where $\beta>0$ is chosen such that $|\Omega| = |B_\beta \setminus \overline{B_\alpha}|$, which can be regarded as a reverse Faber-Krahn type inequality. We establish this result for any $h \in (-\infty,+\infty]$. Moreover, we provide related estimates for higher eigenvalues under additional geometric assumptions on $\Omega$, which can be seen as Szeg\H{o}-Weinberger type inequalities. A few counterexamples to the obtained inequalities for domains violating imposed geometric assumptions are given. As auxiliary information, we investigate shapes of eigenfunctions associated with several eigenvalues $\tau_{i}(B_\beta \setminus \overline{B_\alpha})$ and show that they are nonradial at least for all positive and all sufficiently negative $h$ when $i \in \{2,\ldots,N+2\}$. At the same time, we give numerical evidence that, in the planar case $N=2$, already second eigenfunctions can be radial for some $h<0$. The latter fact provides a simple counterexample to the Payne nodal line conjecture in the case of the mixed boundary conditions.

math.AP↗
arXiv2022

Reverse Faber-Krahn inequalities for Zaremba problems

Let $\Omega$ be a multiply-connected domain in $\mathbb{R}^n$ ($n\geq 2$) of the form $\Omega=\Omega_{\text{out}}\setminus \bar{\Omega_{\text{in}}}.$ Set $\Omega_D$ to be either $\Omega_{\text{out}}$ or $\Omega_{\text{in}}$. For $p\in (1,\infty),$ and $q\in [1,p],$ let $\tau_{1,q}(\Omega)$ be the first eigenvalue of \begin{equation*} -\Delta_p u =\tau \left(\int_{\Omega}|u|^q \text{d}x \right)^{\frac{p-q}{q}} |u|^{q-2}u\;\text{in} \;\Omega,\; u =0\;\text{on}\;\partial\Omega_D, \frac{\partial u}{\partial \eta}=0\;\text{on}\; \partial \Omega\setminus \partial \Omega_D. \end{equation*} Under the assumption that $\Omega_D$ is convex, we establish the following reverse Faber-Krahn inequality $$\tau_{1,q}(\Omega)\leq \tau_{1,q}({\Omega}^\bigstar),$$ where ${\Omega}^\bigstar=B_R\setminus \bar{B_r}$ is a concentric annular region in $\mathbb{R}^n$ having the same Lebesgue measure as $\Omega$ and such that (i) (when $\Omega_D=\Omega_{\text{out}}$) $W_1(\Omega_D)= \omega_n R^{n-1}$, and $(\Omega^\bigstar)_D=B_R$, (ii) (when $\Omega_D=\Omega_{\text{in}}$) $W_{n-1}(\Omega_D)=\omega_nr$, and $(\Omega^\bigstar)_D=B_r$. Here $W_{i}(\Omega_D)$ is the $i^{\text{th}}$ $quermassintegral$ of $\Omega_D.$ We also establish Sz. Nagy's type inequalities for parallel sets of a convex domain in $\mathbb{R}^n$ ($n\geq 3$) for our proof.

math.AP↗
arXiv2022

On the generalised Brezis-Nirenberg problem

For $ p \in (1,N)$ and a domain $Ω$ in $\mathbb{R}^N$, we study the following quasi-linear problem involving the critical growth: \begin{eqnarray*} -Δ_p u - μg|u|^{p-2}u = |u|^{p^{*}-2}u \ \mbox{ in } \mathcal{D}_p(Ω), \end{eqnarray*} where $Δ_p$ is the $p$-Laplace operator defined as $Δ_p(u) = \text{div}(|\nabla u|^{p-2} \nabla u),$ $p^{*}= \frac{Np}{N-p}$ is the critical Sobolev exponent and $\mathcal{D}_p(Ω)$ is the Beppo-Levi space defined as the completion of $\text{C}_c^{\infty}(Ω)$ with respect to the norm $\|u\|_{\mathcal{D}_p} := \left[ \displaystyle \int_Ω |\nabla u|^p \mathrm{d}x \right]^ \frac{1}{p}.$ In this article, we provide various sufficient conditions on $g$ and $Ω$ so that the above problem admits a positive solution for certain range of $μ$. As a consequence, for $N \geq p^2$, if $g $ is such that $g^+ \neq 0$ and the map $u \mapsto \displaystyle \int_Ω |g||u|^p \mathrm{d}x$ is compact on $\mathcal{D}_p(Ω)$, we show that the problem under consideration has a positive solution for certain range of $μ$. Further, for $Ω=\mathbb{R}^N$, we give a necessary condition for the existence of positive solution.

math.AP↗
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↗
arXiv2021

Szegő-Weinberger type inequalities for symmetric domains with holes

Let $μ_2(Ω)$ be the first positive eigenvalue of the Neumann Laplacian in a bounded domain $Ω\subset\mathbb{R}^N$. It was proved by Szegő for $N=2$ and by Weinberger for $N \geq 2$ that among all equimeasurable domains $μ_2(Ω)$ attains its global maximum if $Ω$ is a ball. In the present work, we develop the approach of Weinberger in two directions. Firstly, we refine the Szegő-Weinberger result for a class of domains of the form $Ω_{\text{out}}\setminus\overlineΩ_{\text{in}}$ which are either centrally symmetric or symmetric of order $2$ (with respect to every coordinate plane $(x_i,x_j)$) by showing that $μ_{2}(Ω_{\text{out}}\setminus\overlineΩ_{\text{in}})\leqμ_2(B_β\setminus\overline{B}_α)$, where $B_α, B_β$ are balls centered at the origin such that $B_α\subsetΩ_{\text{in}}$ and $|Ω_{\text{out}}\setminus\overlineΩ_{\text{in}}|=|B_β\setminus\overline{B}_α|$. Secondly, we provide Szegő-Weinberger type inequalities for higher eigenvalues by imposing additional symmetry assumptions on the domain. Namely, if $Ω_{\text{out}}\setminus\overlineΩ_{\text{in}}$ is symmetric of order $4$, then we prove $μ_{i}(Ω_{\text{out}}\setminus\overlineΩ_{\text{in}})\leqμ_i(B_β\setminus\overline{B}_α)$ for $i=3,\dots,N+2$, where we also allow $Ω_{\text{in}}$ and $B_α$ to be empty. If $N=2$ and the domain is symmetric of order $8$, then the latter inequality persists for $i=5$. Counterexamples to the obtained inequalities for domains outside of the considered symmetry classes are given. The existence and properties of nonradial domains with required symmetries in higher dimensions are discussed. As an auxiliary result, we obtain the non-radiality of the eigenfunctions associated to $μ_{N+2}(B_β\setminus\overline{B}_α)$.

math.AP↗
arXiv2021

The compactness and the concentration compactness via $p$-capacity

For $p \in (1,N)$ and $Ω\subseteq \mathbb{R}^N$ open, the Beppo-Levi space $\mathcal{D}^{1,p}_0(Ω)$ is the completion of $C_c^{\infty}(Ω)$ with respect to the norm $\left( \int_Ω|\nabla u|^p \right)^ \frac{1}{p}.$ Using the $p$-capacity, we define a norm and then identify the Banach function space $\mathcal{H}(Ω)$ with the set of all $g$ in $L^1_{loc}(Ω)$ that admits the following Hardy-Sobolev type inequality: \begin{eqnarray*} \int_Ω |g| |u|^p \leq C \int_Ω |\nabla u|^p, \forall\; u \in \mathcal{D}^{1,p}_0(Ω), \end{eqnarray*} for some $C>0.$ Further, we characterize the set of all $g$ in $\mathcal{H}(Ω)$ for which the map $G(u)= \int_Ω g |u|^p$ is compact on $\mathcal{D}^{1,p}_0(Ω)$. We use a variation of the concentration compactness lemma to give a sufficient condition on $g\in \mathcal{H}(Ω)$ so that the best constant in the above inequality is attained in $\mathcal{D}^{1,p}_0(Ω)$.

math.AP↗
arXiv2021

On the Generalized Hardy-Rellich Inequalities

In this article, we look for the weight functions (say $g$) that admits the following generalized Hardy-Rellich type inequality: $ \int_Ω g(x) u^2 dx \leq C \int_Ω |Δu|^2 dx, \forall u \in \mathcal{D}^{2,2}_0(Ω), $ for some constant $C>0$, where $Ω$ is an open set in $\mathbb{R}^N$ with $N\ge 1$. We find various classes of such weight functions, depending on the dimension $N$ and the geometry of $Ω.$ Firstly, we use the Muckenhoupt condition for the one dimensional weighted Hardy inequalities and a symmetrization inequality to obtain admissible weights in certain Lorentz-Zygmund spaces. Secondly, using the fundamental theorem of integration we obtain the weight functions in certain weighted Lebesgue spaces. As a consequence of our results, we obtain simple proofs for the embeddings of $\mathcal{D}^{2,2}_0(Ω)$ into certain Lorentz-Zygmund spaces proved by Hansson and later by Brezis and Wainger.

math.AP↗
arXiv2020

Admissible function spaces for weighted Sobolev inequalities

Let $k,N \in \mathbb{N}$ with $1\le k\le N$ and let $\Omega=\Omega_1 \times \Omega_2$ be an open set in $\mathbb{R}^k \times \mathbb{R}^{N-k}$. For $p\in (1,\infty)$ and $q \in (0,\infty),$ we consider the following Hardy-Sobolev type inequality: \begin{align} \int_{\Omega} |g_1(y)g_2(z)| |u(y,z)|^q \, dy \, dz \leq C \left( \int_{\Omega} | \nabla u(y,z) |^p \, dy \, dz \right)^{\frac{q}{p}}, \quad \forall \, u \in \mathcal{C}^1_c(\Omega), \end{align} for some $C>0$. Depending on the values of $N,k,p,q,$ we have identified various pairs of Lorentz spaces, Lorentz-Zygmund spaces and weighted Lebesgue spaces for $(g_1, g_2)$ so that the above inequality holds. Furthermore, we give a sufficient condition on $g_1,g_2$ so that the best constant in the above inequality is attained in the Beppo-Levi space $\mathcal{D}^{1,p}_0(\Omega)$-the completion of $\mathcal{C}^1_c(\Omega)$ with respect to $\|\nabla u\|_{L^p(\Omega)}$.

math.AP↗
arXiv2020

On global bifurcation for the nonlinear Steklov problems

For $p \in (1, \infty),$ for an integer $N \geq 2$ and for a bounded Lipschitz domain $\Omega$, we consider the following nonlinear Steklov bifurcation problem \begin{equation*} \begin{aligned} -\Delta_p \phi & = 0 \; \text{in} \ \Omega, \\ |\nabla \phi|^{p-2} \frac{\partial \phi}{\partial \nu} &= \lambda \left( g |\phi|^{p-2}\phi + f r(\phi) \right) \; \text{on} \ \partial \Omega, \end{aligned} \end{equation*} where $\Delta_p$ is the $p$-Laplace operator, $g,f \in L^1(\partial \Omega)$ are indefinite weight functions and $r \in C(\mathbb R)$ satisfies $r(0)=0$ and certain growth conditions near zero and at infinity. For $f,g$ in some appropriate Lorentz-Zygmund spaces, we establish the existence of a continuum that bifurcates from $(\lambda_1,0)$, where $\lambda_1$ is the first eigenvalue of the following nonlinear Steklov eigenvalue problem \begin{equation*} \begin{aligned} -\Delta_p \phi & = 0 \; \text{in} \ \Omega, \\ |\nabla \phi|^{p-2} \frac{\partial \phi}{\partial \nu} &= \lambda g |\phi|^{p-2}\phi \ \text{on} \ \partial \Omega. \end{aligned} \end{equation*}

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↗