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Ujjal Das

Publications and source records attributed to Ujjal Das.

At least 19 recordsLinked to original sources
arXiv2026

An optimal fractional Hardy inequality on the discrete half-line

In the context of Hardy inequalities for the fractional Laplacian $(-Δ_{\mathbb{N}})^σ$ on the discrete half-line $\mathbb{N}$, we provide an optimal Hardy-weight $W^{\mathrm{op}}_σ$ for exponents $σ\in\left(0,1\right]$. As a consequence, we provide the sharp constant in the fractional Hardy inequality with the classical Hardy-weight $n^{-2σ}$ on $\mathbb{N}$. It turns out that for $σ=1$ the Hardy-weight $W^{\mathrm{op}}_{1}$ is pointwise larger than the optimal Hardy-weight obtained by Keller--Pinchover--Pogorzelski near infinity. As an application of our main result, we obtain unique continuation results at infinity for the solutions of some fractional Schrödinger equation.

math.AP↗
arXiv2026

On existence of minimizers for weighted $L^p$-Hardy inequalities on $C^{1,γ}$-domains with compact boundary

Let $p \in (1,\infty)$, $α\in \mathbb{R}$, and $Ω\subsetneq \mathbb{R}^N$ be a $C^{1,γ}$-domain with a compact boundary $\partial Ω$, where $γ\in (0,1]$. Denote by $δ_Ω(x)$ the distance of a point $x\in Ω$ to $\partial Ω$. Let $\widetilde{W}^{1,p;α}_0(Ω)$ be the closure of $C_c^{\infty}(Ω)$ in $\widetilde{W}^{1,p;α}(Ω)$, where $$\widetilde{W}^{1,p;α}(Ω):= \left\{φ\in {W}^{1,p}_{\mathrm{loc}} (Ω) \mid \left( \| \, |\nabla φ\, |\|_{L^p(Ω;δ_Ω^{-α})}^p + \|φ\|_{L^p(Ω;δ_Ω^{-(α+p)})}^p\right)<\infty \!\right\}.$$ We study the following two variational constants: the weighted Hardy constant \begin{align*} H_{α,p}(Ω): =\!\inf \left\{\int_Ω |\nabla φ|^p δ_Ω^{-α} \mathrm{d}x \biggm| \int_Ω |φ|^p δ_Ω^{-(α+p)} \mathrm{d}x\!=\!1, φ\in \widetilde{W}^{1,p;α}_0(Ω) \right\} , \end{align*} and the weighted Hardy constant at infinity \begin{align*} λ_{α,p}^{\infty}(Ω) :=\sup_{K\Subset Ω}\, \inf_{W^{1,p}_{c}(Ω\setminus \overline{K})} \left\{\int_{Ω\setminus \overline{K}} |\nabla φ|^p δ_Ω^{-α} \mathrm{d}x \biggm| \int_{Ω\setminus \overline{K}} |φ|^p δ_Ω^{-(α+p)} \mathrm{d}x=1 \right\}. \end{align*} We show that $H_{α,p}(Ω)$ is attained if and only if the spectral gap $Γ_{α,p}(Ω):= λ_{α,p}^{\infty}(Ω)-H_{α,p}(Ω)$ is strictly positive. Moreover, we obtain tight decay estimates for the corresponding minimizers.

math.AP↗
arXiv2026

Characterizations of compactness and weighted eigenvalue problem associated with fractional Hardy-type inequalities

In this article, we consider the following fractional {Hardy-type} inequality: \begin{align} \label{Fractional Hardy_abst} \int_{\mathbb{R}^N} |w(x)||u(x)|^p \mathrm{d}x \leq C \int_{\mathbb{R}^N \times \mathbb{R}^N} \frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}} \mathrm{d}x\mathrm{d}y:= \|u\|_{s,p}^p\,, \ \forall u \in \mathcal{D}^{s,p}(\mathbb{R}^N), \end{align} where $0<s<1<p<\frac{N}{s}$, and $\mathcal{D}^{s,p}(\mathbb{R}^N)$ is the completion of $C_c^1(\mathbb{R}^N)$ with respect to the {norm} $\|\cdot\|_{s,p}$. We denote the space of admissible {weight function} $w$ in \eqref{Fractional Hardy_abst} by $\mathcal{H}_{s,p}(\mathbb{R}^N)$. Maz'ya-type characterization helps us to define a Banach function norm on $\mathcal{H}_{s,p}(\mathbb{R}^N)$. Using the Banach function space structure and the concentration compactness type arguments, we provide several characterizations for the compactness of the map ${W}(u)= \int_{{\mathbb{R}^N}} |w| |u|^p \mathrm{d}x$ on $\mathcal{D}^{s,p}(\mathbb{R}^N)$. In particular, we prove that ${W}$ is compact on $\mathcal{D}^{s,p}(\mathbb{R}^N)$ if and only if $w \in \mathcal{H}_{s,p,0}(\mathbb{R}^N):=\overline{C_c(\mathbb{R}^N)} \ \mbox{in} \ \mathcal{H}_{s,p}(\mathbb{R}^N)$. Further, we study the following {weighted} eigenvalue problem: \begin{equation*} (-Δ_{p})^{s}u = λw(x) |u|^{p-2}u ~~\text{in}~\mathbb{R}^{N}, \end{equation*} where $(-Δ_{p})^{s}$ is the fractional $p$-Laplace operator and $w = w_{1} - w_{2}~\text{with}~ w_{1},w_{2} \geq 0,$ is such that $ w_{1} \in \mathcal{H}_{s,p,0}(\mathbb{R}^N)$ and $w_{2} \in L^{1}_{loc}(\mathbb{R}^N)$.

math.AP↗
arXiv2026

Quantitative Landis-type result for Dirac operators

We study quantitative unique continuation at infinity for Dirac equations with bounded matrix-valued potentials. For the massless Dirac operator $\mathcal{D}_n$ in $\mathbb{R}^n$, we establish a Landis-type estimate showing that the vanishing order of any nontrivial bounded solution of $( \mathcal{D}_n + \mathbb{V} ) φ= 0$ satisfies a lower bound of order $\exp(-κR^{2} (\log R)^{2})$ as $|x|=R\to \infty$; the quadratic growth in the exponent is sharp, in view of previous known results. Our proof follows a Bourgain--Kenig type approach based on a Carleman inequality for Dirac operators which relies on a local Hölder regularity result, which we also prove. In two dimension, we obtain improved quantitative estimates under symmetry assumptions on the potential $\mathbb{V}$ and for real-valued solutions. Finally, we also derive qualitative Landis-type results for Dirac equations with decaying potentials, including critical decay rates.

math.AP↗
arXiv2025

On the Landis Conjecture for Positive Quasi-linear Operators on Graphs

We prove a Landis type unique continuation result for positive quasi-linear operators on graphs. Specifically, we give decay criteria that ensures when a harmonic function for a positive quasilinear Schrödinger operator with potential less than 1 is trivially zero. The assumption of positivity of the operator allows the application of criticality theory such as the Liouville comparison theorem. Furthermore, our results fundamentally build on the so called simplified energy. As an application we discuss the case of model graphs and in particular regular trees.

math.AP↗
arXiv2025

On Landis' conjecture for positive Schrödinger operators on graphs

In this note we study the Landis conjecture for positive Schrödin\-ger operators on graphs. More precisely, we prove a Landis-type result in the form of a decay criterion that ensures when $\mathcal{H}$-harmonic functions for a positive Schrödinger operator $\mathcal{H}$ with potentials bounded from above by $ 1 $ are trivial. The positivity assumption on the operator allows us to impose slow decay across the entire graph, while requiring fast decay in only one direction, rather than throughout the whole graph. We then specifically look at the special cases of $ \mathbb{Z}^{d} $ and regular trees for which we get a explicit decay criterion. Moreover, we consider the fractional analogue of the Landis conjecture on $ \mathbb{Z}^{d} $. Our approach relies on the discrete version of Liouville comparison principle which is also proved in this article.

math.AP↗
arXiv2025

Information Extraction from Visually Rich Documents using LLM-based Organization of Documents into Independent Textual Segments

Information extraction (IE) from Visually Rich Documents (VRDs) containing layout features along with text is a critical and well-studied task. Specialized non-LLM NLP-based solutions typically involve training models using both textual and geometric information to label sequences/tokens as named entities or answers to specific questions. However, these approaches lack reasoning, are not able to infer values not explicitly present in documents, and do not generalize well to new formats. Generative LLM-based approaches proposed recently are capable of reasoning, but struggle to comprehend clues from document layout especially in previously unseen document formats, and do not show competitive performance in heterogeneous VRD benchmark datasets. In this paper, we propose BLOCKIE, a novel LLM-based approach that organizes VRDs into localized, reusable semantic textual segments called $\textit{semantic blocks}$, which are processed independently. Through focused and more generalizable reasoning,our approach outperforms the state-of-the-art on public VRD benchmarks by 1-3% in F1 scores, is resilient to document formats previously not encountered and shows abilities to correctly extract information not explicitly present in documents.

cs.IR↗
arXiv2024

The space of Hardy-weights for quasilinear operators on discrete graphs

We study Hardy inequalities for $p$-Schrödinger operators on general weighted graphs. Specifically, we prove a Maz'ya-type result, where we characterize the space of Hardy weights for $ p $-Schrödinger operators via a generalized capacity. The novel ingredient in the proof is the demonstration that the simplified energy of the $ p $-Schrödinger energy functional is compatible with certain normal contractions. As a consequence, we obtain a necessary integrability criterion for Hardy weights. Finally, using some tools of criticality theory, we investigate the existence of minimizers in the Hardy inequalities and discuss relations to Cheeger type estimates.

math.AP↗
arXiv2024

The Landis conjecture via Liouville comparison principle and criticality theory

We give partial affirmative answers to Landis conjecture in all dimensions for two different types of linear, second order, elliptic operators in a domain $Ω\subset \mathbb{R}^N$. In particular, we provide a sharp decay criterion that ensures when a solution of a nonnegative Schrödinger equation in $\mathbb{R}^N$ with a potential $V\leq 1$ is trivial. Moreover, we address the analogue of Landis conjecture for quasilinear problems. Our approach relies on the application of Liouville comparison principles and criticality theory.

math.AP↗
arXiv2023

On Weighted Orlicz-Sobolev inequalities

Let $Ω$ be an open subset of $\mathbb{R}^N$ with $N\geq 2.$ We identify various classes of Young functions $Φ$ and $Ψ$, and function spaces for a weight function $g$ so that the following weighted Orlicz-Sobolev inequality holds: \begin{equation*}\label{ineq:Orlicz} Ψ^{-1}\left(\int_Ω|g(x)|\,Ψ(|u(x)| )dx \right)\leq CΦ^{-1}\left(\int_ΩΦ(|\nabla u(x)|) dx \right),\;\;\;\forall\,u\in \mathcal{C}^1_c(Ω), \end{equation*} for some $C>0$. As an application, we study the existence of eigenvalues for certain nonlinear weighted eigenvalue problems.

math.AP↗
arXiv2022

Contrasting Analog and Digital Resistive Switching Memory Characteristics in Solution-Processed Copper (I) Thiocyanate and Its Polymer Electrolyte Based Memristive Devices

Usually, resistive switching (RS) devices show digital RS memory (sharp SET and RESET process), which is most suitable for digital data storage applications. Some RS devices also manifest ideal memristive behavior or analog memory characteristics (gradual change in resistance states). The analog RS properties of memristive devices widen their application domain to a much broader field of neuromorphic computing. The tunability of memristive devices to digital or analog memory applications greatly depends upon the switching medium. In this work, we report a comparative study on RS properties of two kinds of memristive devices based upon copper (I) thiocyanate (CuSCN) and a solid polymer electrolyte (SPE) made up of CuSCN as ionic moieties in polyethylene oxide (PEO). The device (ITO/CuSCN/Cu), prepared by spin-coating CuSCN layer between ITO and copper electrode, shows simultaneous analog and digital RS characteristics. The RS property of the device is tunable by varying the thickness of the CuSCN layer. The current-voltage characteristics reveal that devices prepared at 3000 rpm (thicker) during the spin-coating show only digital bipolar RS memory. In comparison, the devices deposited at 4000 rpm (thinner) show both analog and digital RS memory. The conduction mechanism responsible for RS behavior in CuSCN-based devices is Schottky emission mediated charge trapping and de-trapping at the interfacial states. Contrastingly, when the same CuSCN is used as the electrolyte in SPE film, the device only shows bipolar digital non-volatile memory characteristics. The RS behavior is due to the electrochemical metallization (ECM) mechanism. The ON and OFF states are achieved by the formation and rupture of copper filaments due to the redox reactions at the interface.

cond-mat.mes-hall↗
arXiv2022

Admissible function spaces for weighted Sobolev inequalities

Let $k,N \in \mathbb{N}$ with $1\le k\le N$ and let $Ω=Ω_1 \times Ω_2$ 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_Ω |g_1(y)g_2(z)| |u(y,z)|^q \, dy \, dz \leq C \left( \int_Ω | \nabla u(y,z) |^p \, dy \, dz \right)^{\frac{q}{p}}, \quad \forall \, u \in \mathcal{C}^1_c(Ω), \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 $\mathcal{D}^{1,p}_0(Ω)$-the completion of $\mathcal{C}^1_c(Ω)$ with respect to $\|\nabla u\|_{L^p(Ω)}$.

math.AP↗
arXiv2022

On the optimization of the first weighted eigenvalue

For $N\geq 2$, a bounded smooth domain $Ω$ in $\mathbb{R}^N$, and $g_0, V_0 \in L^1_{loc}(Ω)$, we study the optimization of the first eigenvalue for the following weighted eigenvalue problem: \begin{align*} -Δ_p ϕ+ V |ϕ|^{p-2}ϕ= λg |ϕ|^{p-2}ϕ\text{ in } Ω, \quad ϕ=0 \text{ on } \partial Ω, \end{align*} where $g$ and $V$ vary over the rearrangement classes of $g_0$ and $V_0$, respectively. We prove the existence of a minimizing pair $(\underline{g},\underline{V})$ and a maximizing pair $(\overline{g},\overline{V})$ for $g_0$ and $V_0$ lying in certain Lebesgue spaces. We obtain various qualitative properties such as polarization invariance, Steiner symmetry of the minimizers as well as the associated eigenfunctions for the case $p=2$. For annular domains, we prove that the minimizers and the corresponding eigenfunctions possess the foliated Schwarz symmetry.

math.AP↗
arXiv2022

On the fourth order semipositone problem in $\mathbb{R}^N$

For $N \geq 5$ and $a>0$, we consider the following semipositone problem \begin{align*} Δ^2 u= g(x)f_a(u) \text { in } \mathbb{R}^N, \, \text{ and } \, u \in \mathcal{D}^{2,2}(\mathbb{R}^N),\ \ \ \qquad \quad \mathrm{(SP)} \end{align*} where $g \in L^1_{loc}(\mathbb{R}^N)$ is an indefinite weight function, $f_a:\mathbb{R} \to \mathbb{R}$ is a continuous function that satisfies $f_a(t)=-a$ for $t \in \mathbb{R}^-$, and $\mathcal{D}^{2,2}(\mathbb{R}^N)$ is the completion of $\mathcal{C}_c^{\infty}(\mathbb{R}^N)$ with respect to $(\int_{\mathbb{R}^N} (Δu)^2)^{1/2}$. For $f_a$ satisfying subcritical nonlinearity and a weaker Ambrosetti-Rabinowitz type growth condition, we find the existence of $a_1>0$ such that for each $a \in (0,a_1)$, (SP) admits a mountain pass solution. Further, we show that the mountain pass solution is positive if $a$ is near zero. For the positivity, we derive uniform regularity estimates of the solutions of (SP) for certain ranges in $(0,a_1)$, relying on the Riesz potential of the biharmonic operator.

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

The space of Hardy-weights for quasilinear equations: Maz'ya-type characterization and sufficient conditions for existence of minimizers

Let $p \in (1,\infty)$ and $Ω\subset \mathbb{R}^N$ be a domain. Let $ A: =(a_{ij}) \in L^{\infty}_{\text{loc}}(Ω; \mathbb{R}^{N\times N})$ be a symmetric and locally uniformly positive definite matrix. Set $|ξ|_A^2:= \displaystyle \sum_{i,j=1}^N a_{ij}(x) ξ_i ξ_j$, $ξ\in \mathbb{R}^N$, and let $V$ be a given potential in a certain local Morrey space. We assume that the energy functional $$Q_{p,A,V}(ϕ):=\displaystyle \int_Ω [|\nabla ϕ|_A^p + V|ϕ|^p] {\rm dx} $$ is nonnegative in $W^{1,p}(Ω)\cap C_c(Ω)$. We introduce a generalized notion of $Q_{p,A,V}$-capacity and characterize the space of all Hardy-weights for the functional $Q_{p,A,V}$, extending Maz'ya's well known characterization of the space of Hardy-weights for the $p$-Laplacian. In addition, we provide various sufficient conditions on the potential $V$ and the Hardy-weight $g$ such that the best constant of the corresponding variational problem is attained in an appropriate Beppo-Levi space.

math.AP↗