arXiv subjects

Sallah Eddine Boutiah

Publications and source records attributed to Sallah Eddine Boutiah.

3 recordsLinked to original sources

arXiv2026

The $L^1$-realization of an elliptic operator with unbounded coefficients

We establish an $L^1$-theory for a class of elliptic operators with unbounded coefficients, extending the semigroup generation results of Boutiah et al. (J. Differential Equations 264 (2018), 2184--2204) and Canale et al. (Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16 (2016), 581--601) to the endpoint $p=1$. The $L^1$-realization generates a positive analytic $C_0$-semigroup that is compact for every positive time. We characterize the generator domain through weighted estimates and prove consistency with the corresponding $L^p$-semigroups for $1<p<\infty$.

math.AP↗

arXiv2026

Lower bounds for heat kernels of elliptic operators with unbounded diffusion, drift, and potential terms

We establish a pointwise lower bound for the heat kernel of the elliptic operator $ Λ=(1+|x|^α)Δ+b|x|^{α-2}x\cdot\nabla-|x|^β$, where $d\geq3$, $α>2$, $β>α-2$, and $b\in\mathbb R$. For every $τ>0$, we prove that $$ \begin{aligned} p(t,x,y) &\geq C_τe^{λ_0t} \left(\frac{1+|y|^α}{1+|x|^α}\right)^{\frac{b}{2α}} \frac{(|x||y|)^{-\frac{d-1}{2}-\frac{β-α}{4}}}{1+|y|^α}\\ &\quad\times\exp \left[ -\int_1^{|x|}\sqrt{\frac{s^β}{1+s^α}}\,\mathrm ds -\int_1^{|y|}\sqrt{\frac{s^β}{1+s^α}}\,\mathrm ds \right] \end{aligned} $$ for all $t\geqτ$ and $|x|,|y|\geq1$, where $λ_0<0$ is the largest eigenvalue of $Λ$ and $C_τ>0$ is independent of $t,x,y$. The proof applies the classical Davies - Simon argument in a weighted symmetric setting.

math.AP↗

arXiv2021

Some results on second-order elliptic operators with polynomially growing coefficients in $L^p$-spaces

In this paper we study minimal realizations in $L^p(\mathbb{R}^N)$ of the second order elliptic operator \begin{equation*} { A_{b,c}} := (1+|x|^α)Δ+ b|x|^{α-2}x\cdot\nabla - c |x|^{α-2} - |x|^β , \quad x \in \mathbb{R}^N, \end{equation*} where $N\geq3$, $α\in[0,2)$, $β>0$, and $b, c$ are real numbers. We use quadratic form methods to prove that $\left(A_{b,c},C_c^\infty\left(\mathbb{R}^N\setminus \{0\}\right)\right)$ admits an extension that generates an analytic $C_0-$semigroup for all $p\in(1,\infty)$. Moreover, we give conditions on the coefficients under which this extension is precisely the closure of $\left(A_{b,c},C_c^\infty\left(\mathbb{R}^N\setminus \{0\}\right)\right)$.

math.AP↗