Searcharxiv⌕ Search

arXiv subjects

Debajyoti Choudhuri

Publications and source records attributed to Debajyoti Choudhuri.

31 records · Page 2Linked to original sources
arXiv2021

Self-duality of One-dimensional Quasicrystals with Spin-Orbit Interaction

Non-interacting spinless electrons in one-dimensional quasicrystals, described by the Aubry-André-Harper (AAH) Hamiltonian with nearest neighbour hopping, undergoes metal to insulator transition (MIT) at a critical strength of the quasi-periodic potential. This transition is related to the self-duality of the AAH Hamiltonian. Interestingly, at the critical point, which is also known as the self-dual point, all the single particle wave functions are multifractal or non-ergodic in nature, while they are ergodic and delocalized (localized) below (above) the critical point. In this work, we have studied the one dimensional quasi-periodic AAH Hamiltonian in the presence of spin-orbit (SO) coupling of Rashba type, which introduces an additional spin conserving complex hopping and a spin-flip hopping. We have found that, although the self-dual nature of AAH Hamiltonian remains unaltered, the self-dual point gets affected significantly. Moreover, the effect of the complex and spin-flip hoppings are identical in nature. We have extended the idea of Kohn's localization tensor calculations for quasi-particles and detected the critical point very accurately. These calculations are followed by detailed multifractal analysis along with the computation of inverse participation ratio and von Neumann entropy, which clearly demonstrate that the quasi-particle eigenstates are indeed multifractal and non-ergodic at the critical point. Finally, we mapped out the phase diagram in the parameter space of quasi-periodic potential and SO coupling strength.

cond-mat.dis-nn↗
arXiv2020

Existence and Hölder regularity of infinitely many solutions to a $p$-Kirchhoff type problem involving a singular nonlinearity without the Ambrosetti-Rabinowitz (AR) condition

We carry out an investigation of the existence of infinitely many solutions to a fractional $p$-Kirchhoff type problem with a singularity and a superlinear nonlinearity with a homogeneous Dirichlet boundary condition. Further the solution(s) will be proved to be bounded and a weak comparison principle has also been proved. A {\it `$C^1$ versus $W_0^{s,p}$'} analysis has also been discussed.

math.AP↗
arXiv2020

Existence of positive solutions for a singular elliptic problem with critical exponent and measure data

We prove the existence of a positive {\it SOLA (Solutions Obtained as Limits of Approximations)} to the following PDE involving fractional power of Laplacian \begin{equation} \begin{split} (-Δ)^su&= \frac{1}{u^γ}+λu^{2_s^*-1}+μ~\text{in}~Ω, u&>0~\text{in}~Ω, u&= 0~\text{in}~\mathbb{R}^N\setminusΩ. \end{split} \end{equation} Here, $Ω$ is a bounded domain of $\mathbb{R}^N$, $s\in (0,1)$, $2s<N$, $λ,γ\in (0,1)$, $2_s^*=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent and $μ$ is a nonnegative bounded Radon measure in $Ω$.

math.AP↗
arXiv2020

Infinitely many small solutions to an elliptic PDE of variable exponent with a singular nonlinearity

We prove the existence of infinitely many nonnegative solutions to the following nonlocal elliptic partial differential equation involving singularities \begin{align} (-Δ)_{p(\cdot)}^{s} u&=\fracλ{|u|^{γ(x)-1}u}+f(x,u)~\text{in}~Ω,\nonumber u&=0~\text{in}~\mathbb{R}^N\setminusΩ,\nonumber \end{align} where $Ω\subset\mathbb{R}^N,\, N\geq2$ is a smooth, bounded domain, $λ>0$, $s\in (0,1)$, $γ(x)\in(0,1)$ for all $x\in\barΩ$, $N>sp(x,y)$ for all $(x,y)\in\barΩ\times\barΩ$ and $(-Δ)_{p(\cdot)}^{s}$ is the fractional $p(\cdot)$-Laplacian operator with variable exponent. The nonlinear function $f$ satisfies certain growth conditions. Moreover, we establish a uniform $L^{\infty}(\barΩ)$ estimate of the solution(s) by the Moser iteration technique.

math.AP↗
arXiv2020

Existence and multiplicity of solutions to a $p-q$ Laplacian system with a concave and singular nonlinearities

In this paper we study the existence of multiple nontrivial positive weak solutions to the following system of problems. \begin{align*} \begin{split} -Δ_{p}u-Δ_q u &= λf(x)|u|^{r-2}u+ν\frac{1-α}{2-α-β}h(x) |u|^{-α}|v|^{1-β}\,\,\mbox{in}\,\,Ω,\\ -Δ_{p}v-Δ_q v &= μg(x)|v|^{r-2}v+ν\frac{1-β}{2-α-β}h(x) |u|^{1-α}|v|^{-β}\,\,\mbox{in}\,\,Ω,\\ u,v&>0\,\,\mbox{in}\,\,Ω,\\ u= v &= 0\,\, \mbox{on}\,\, \partialΩ\end{split} \end{align*} where (C):~$0<α<1,\;0<β<1,$ $2-α-β<q<\frac{N(p-1)}{N-p}<p<r<p^*$, with $p^*=\frac{Np}{N-p}$. We will guarantee the existence of a solution in the Nehari manifold. Further by using the Lusternik-Schnirelman category we will prove the existence of at least $\text{cat}(Ω)+1$ number of solutions.

math.AP↗
arXiv2019

Existence results of two mixed boundary value elliptic PDEs in $\mathbb{R}^n$

We study the existence of a solution to the mixed boundary value problem for Helmholtz and Poisson type equations in a bounded Lipschitz domain $Ω\subset\mathbb{R}^N$ and in $\mathbb{R}^N\setminusΩ$ for $N\geq3$. The boundary $\partialΩ$ of $Ω$ is the decomposition of $Γ_1,Γ_2\subset\partialΩ$ such that $\partialΩ=Γ=\overlineΓ_1\cupΓ_2=Γ_1\cup\overlineΓ_2$ and $Γ_1\capΓ_2=\emptyset$. We have shown that if the Neumann data $f_2\in H^{-\frac{1}{2}}(Γ_2)$ and the Dirichlet data $f_1\in H^{\frac{1}{2}}(Γ_1)$ then the Helmholtz problem with mixed boundary data admits a unique solution. We have also shown the existence of a weak solution to a mixed boundary value problem governed by the Poisson equation with a measure data and the Dirichlet, Neumann data belongs to $f_1\in H^{\frac{1}{2}}(Γ_1)$, $f_2\in H^{-\frac{1}{2}}(Γ_2)$ respectively.

math.AP↗
arXiv2019

Multiplicity and Hölder regularity of solutions for a nonlocal elliptic PDE involving singularity

In this paper, we prove the existence of multiple solutions for a nonlinear nonlocal elliptic PDE involving a singularity which is given as \begin{eqnarray} (-Δ_p)^s u&=& \fracλ{u^γ}+u^q~\text{in}~Ω,\nonumber u&=&0~\text{in}~\mathbb{R}^N\setminusΩ,\nonumber u&>& 0~\text{in}~Ω\nonumber, \end{eqnarray} where $Ω$ is an open bounded domain in $\mathbb{R}^N$ with smooth boundary, $N>ps$, $s\in (0,1)$, $λ>0$, $0<γ<1$, $1<p<\infty$, $p-1<q\leq p_s^{*}=\frac{Np}{N-ps}$. We employ variational techniques to show the existence of multiple positive weak solutions of the above problem. We also prove that for some $β\in (0,1)$, the weak solution to the problem is in $C^{1,β}(\overlineΩ)$.

math.AP↗
arXiv2018

A study of second order semilinear elliptic PDE involving measures

The objective of this article is to study the boundary value problem for the general semilinear elliptic equation of second order involving $L^1$ functions or Radon measures with finite total variation. The study investigates the existence and uniqueness of `{\it very weak}' solutions to the boundary value problem for a given $L^1$ function. However, a `{\it very weak}' solution need not exist when an $L^1$ function is replaced with a measure due to which the corresponding reduced limits has been found for which the problem admits a solution in a `{\it very weak}' sense.

math.AP↗
arXiv2018

Sharp quadrature error bounds for the nearest-neighbor discretization of the regularized stokeslet boundary integral equation

The method of regularized stokeslets is a powerful numerical method to solve the Stokes flow equations for problems in biological fluid mechanics. A recent variation of this method incorporates a nearest-neighbor discretization to improve accuracy and efficiency while maintaining the ease-of-implementation of the original meshless method. This method contains three sources of numerical error, the regularization error associated from using the regularized form of the boundary integral equations (with parameter $\varepsilon$), and two sources of discretization error associated with the force and quadrature discretizations (with lengthscales $h_f$ and $h_q$). A key issue to address is the quadrature error: initial work has not fully explained observed numerical convergence phenomena. In the present manuscript we construct sharp quadrature error bounds for the nearest-neighbor discretisation, noting that the error for a single evaluation of the kernel depends on the smallest distance ($δ$) between these discretization sets. The quadrature error bounds are described for two cases: with disjoint sets ($δ>0$) being close to linear in $h_q$ and insensitive to $\varepsilon$, and contained sets ($δ=0$) being quadratic in $h_q$ with inverse dependence on $\varepsilon$. The practical implications of these error bounds are discussed with reference to the condition number of the matrix system for the nearest-neighbor method, with the analysis revealing that the condition number is insensitive to $\varepsilon$ for disjoint sets, and grows linearly with $\varepsilon$ for contained sets. Error bounds for the general case ($δ\geq 0$) are revealed to be proportional to the sum of the errors for each case.

physics.flu-dyn↗
arXiv2017

Existence and Concentration Solutions for a class of elliptic PDEs involving $p$-biharmonic Operator

In this paper, we propose an existence result pertaining to a nontrivial solution to the problem \begin{align*} \Bigg\{\begin{split} & Δ^2_p u -Δ_p u + λV(x)|u|^{p-2}u = f(x,u)\,,\,x\in \mathbb{R}^N, & u \in W^{2,p}(\mathbb{R}^N), \end{split} \end{align*} where $λ>0$, $p>1, N>2p$ and $V\in C(\mathbb{R}^N, \mathbb{R}^+)$, $f\in C(\mathbb{R}^N \times \mathbb{R},\mathbb{R})$ with certain properties. We also investigate the concentration of solutions to the problem on the set $V^{-1}(0)$ as $λ\rightarrow \infty$.

math.AP↗
arXiv2016

Composition operators on Orlicz-Sobolev spaces

The kernel of composition operator $C_T$ on Orlicz-Sobolev space is obtained. Using the kernel, a necessary and a sufficient condition for injectivity of composition operator $C_T$ has been established. Composition operators on Orlicz-Sobolev space with finite ascent as well as infinite ascent have been characterized.

math.FA↗