Searcharxiv⌕ Search

arXiv subjects

Abdelaziz Rhandi

Publications and source records attributed to Abdelaziz Rhandi.

29 records · Page 2Linked to original sources
arXiv2017

Weighted Hardy's inequalities and Kolmogorov-type operators

We give general conditions to state the weighted Hardy inequality \[ c\int_{\mathbb{R}^N}\frac{φ^2} {|x|^2}dμ\leq\int_{\mathbb{R}^N}|\nabla φ|^2 dμ+C\int_{\mathbb{R}^N} φ^2dμ,\quad φ\in C_c^{\infty}(\mathbb{R}^N),\,c\leq c_{0,μ}, \] with respect to a probability measure $dμ$. Moreover, the optimality of the constant $c_{0,μ}$ is given. The inequality is related to the following Kolmogorov equation perturbed by a singular potential \[ Lu+Vu=\left(Δu+\frac{\nabla μ}μ\cdot \nabla u\right)+\frac{c}{|x|^2}u \] for which the existence of positive solutions to the corresponding parabolic problem can be investigated. The hypotheses on $dμ$ allow the drift term to be of type $\frac{\nabla μ}μ= -|x|^{m-2}x$ with $m> 0$.

math.AP↗
arXiv2017

${L^p}$-theory for Schrödinger systems

In this article we study for $p\in (1,\infty)$ the $L^p$-realization of the vector-valued Schrödinger operator $\mathcal{L}u := \mathrm{div} (Q\nabla u) + V u$. Using a noncommutative version of the Dore-Venni theorem due to Monniaux and Prüss, we prove that the $L^p$-realization of $\mathcal{L}$, defined on the intersection of the natural domains of the differential and multiplication operators which form $\mathcal{L}$, generates a strongly continuous contraction semigroup on $L^p(\mathbb{R}^d; \mathbb{R}^m)$. We also study additional properties of the semigroup such as extension to $L^1$, positivity, ultracontractivity and prove that the generator has compact resolvent.

math.AP↗
arXiv2016

Kernel estimates for Schrödinger type operators with unbounded diffusion and potential terms

We prove that the heat kernel associated to the Schrödinger type operator $A:=(1+|x|^α)Δ-|x|^β$ satisfies the estimate $$k(t,x,y)\leq c_1e^{λ_0t}e^{c_2t^{-b}}\frac{(|x||y|)^{-\frac{N-1}{2}-\frac{β-α}{4}}}{1+|y|^α} e^{-\frac{2}{β-α+2}|x|^{\frac{β-α+2}{2}}} e^{-\frac{2}{β-α+2}|y|^{\frac{β-α+2}{2}}} $$ for $t>0,|x|,|y|\ge 1$, where $c_1,c_2$ are positive constants and $b=\frac{β-α+2}{β+α-2}$ provided that $N>2,\,α\geq 2$ and $β>α-2$. We also obtain an estimate of the eigenfunctions of $A$.

math.AP↗
arXiv2016

Elliptic operators with unbounded diffusion coefficients perturbed by inverse square potentials in $L^p$-spaces

In this paper we give sufficient conditions on $α\geq 0$ and $c\in \mathbb{R}$ ensuring that the space of test functions $C_c^\infty(\mathbb{R}^N)$ is a core for the operator $$L_0u=(1+|x|^α)Δu+\frac{c}{|x|^2}u=:Lu+\frac{c}{|x|^2}u,$$ and $L_0$ with suitable domain generates a quasi-contractive and positivity preserving $C_0$-semigroup in $L^p(\mathbb{R}^N),\,1<p<\infty$. The proofs are based on some $L^p$-weighted Hardy's inequality and perturbation techniques.

math.AP↗
arXiv2016

A multispecies birth-death-immigration process and its diffusion approximation

We consider an extended birth-death-immigration process defined on a lattice formed by the integers of $d$ semiaxes joined at the origin. When the process reaches the origin, then it may jumps toward any semiaxis with the same rate. The dynamics on each ray evolves according to a one-dimensional linear birth-death process with immigration. We investigate the transient and asymptotic behavior of the process via its probability generating function. The stationary distribution, when existing, is a zero-modified negative binomial distribution. We also study a diffusive approximation of the process, which involves a diffusion process with linear drift and infinitesimal variance on each ray. It possesses a gamma-type transient density admitting a stationary limit. As a byproduct of our study, we obtain a closed form of the number of permutations with a fixed number of components, and a new series form of the polylogarithm function expressed in terms of the Gauss hypergeometric function.

math.PR↗
arXiv2012

New weighted Hardy's inequalities with application to non-existence of global solutions

In this article, we prove a weighted Hardy inequality for $1 d$, then we can deduce from our weighted Hardy inequality a Poincaré inequality. The proof of the weighted Hardy inequality is based on the method of vector fields firstly introduced by Mitidieri \cite{MR1769903}. By the same method, we show for $1<p<+\infty$ and $d\ge1$ that weighted Caffarelli-Kohn-Nirenberg inequalities hold true. As an application of our weighted Hardy inequality, we prove a non-existence result for a p-Kolmogorov parabolic equation.

math.AP↗
arXiv2012

On Schroedinger type operators with unbounded coefficients: Generation and heat kernel estimates

We consider the Schrödinger type operator ${\mathcal A}=(1+|x|^α)Δ-|x|^β$, for $α\in [0,2]$ and $β\ge 0$. We prove that, for any $p\in (1,\infty)$, the minimal realization of operator ${\mathcal A}$ in $L^p(\R^N)$ generates a strongly continuous analytic semigroup $(T_p(t))_{t\ge 0}$. For $α\in [0,2)$ and $β\ge 2$, we then prove some upper estimates for the heat kernel $k$ associated to the semigroup $(T_p(t))_{t\ge 0}$. As a consequence we obtain an estimate for large $|x|$ of the eigenfunctions of ${\mathcal A}$. Finally, we extend such estimates to a class of divergence type elliptic operators.

math.AP↗
arXiv2010

The Oseen-Navier-Stokes flow in the exterior of a rotating obstacle: The non-autonomous case

Consider the Navier-Stokes flow past a rotating obstacle with a general time-dependent angular velocity and a time-dependent outflow condition at infinity -- sometimes called an Oseen condition. By a suitable change of coordinates the problem is transformed to an non-autonomous problem with unbounded drift terms on a fixed exterior domain $Ω\subset \R^d$. It is shown that the solution to the linearized problem is governed by a strongly continuous evolution system $\{T_Ω(t,s)\}_{t\geq s\geq0}$ on $L^p_σ(Ω)$ for $1<p<\infty$. Moreover, $L^p$-$L^q$ smoothing properties and gradient estimates of $T_Ω(t,s)$, $0\leq s \leq t$, are obtained. These results are the key ingredients to show local in time existence of mild solutions to the full nonlinear problem for $p\geq d$ and initial value in $L^p_σ(Ω)$.

math.AP↗
arXiv2010

Non-autonomous Ornstein-Uhlenbeck equations in exterior domains

In this paper, we consider non-autonomous Ornstein-Uhlenbeck operators in smooth exterior domains $Ω\subset \R^d$ subject to Dirichlet boundary conditions. Under suitable assumptions on the coefficients, the solution of the corresponding non-autonomous parabolic Cauchy problem is governed by an evolution system $\{P_Ω(t,s)\}_{0\leq s\leq t}$ on $L^p(Ω)$ for $1< p < \infty$. Furthermore, $L^p$-estimates for spatial derivatives and $L^p$-$L^q$ smoothing properties of $P_Ω(t,s)$, $0\leq s\leq t$, are obtained.

math.AP↗