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Mohamed Ben Chrouda

Publications and source records attributed to Mohamed Ben Chrouda.

5 recordsLinked to original sources
arXiv2015

Blow up boundary solutions of some semilinear fractional equations in the unit ball

For $γ>0$, we are interested in blow up solutions $u\in C^+(B)$ of the fractional problem in the unit ball $B$ \begin{equation}\label{2nov} \left\{\begin{array} {rcll} Δ^{\fracα{2}} u &=& u^γ&\ \text{in }B\\ u &=& 0&\ \text{in }B^c.\end{array}\right. \end{equation} We distinguish particularly two orders of singularity at the boundary: solutions exploding at the same rate than $δ^{1-\fracα{2}}$ ($δ$ denotes the Euclidean distance) and those higher singular than $δ^{1-\fracα{2}}.$ As a consequence, it will be shown that the classical Keller-Osserman condition can not be readopted in the fractional setting.

math.AP↗
arXiv2015

Semilinear equations associated with Dunkl Laplacian

Let $Δ_k$ be the Dunkl Laplacian on $\mathbb R^d$ associated with a reflection group $W$ and a multiplicity function $k$. The purpose of this paper is to establish necessary and sufficient condition under which there exists a positive solution of the equation $Δ_ku=φ(u)$ in the unit ball of $\mathbb R^d$ as well as in the whole space $\mathbb R^d$.

math.AP↗
arXiv2014

On the Dirichlet problem associated with Dunkl Laplacian

This paper is devoted to the study of the Dirichlet problem associated with the Dunkl Laplacian $Δ_k$. We establish, under some condition on a bounded domain $D$ of $\R^d$, the existence of a unique continuous function $h$ on $\R^d$ such that $Δ_kh=0$ on $D$ and $h=f$ on $\R^d\setminus D$ the complement of $D$ in $\R^d$, where the function $f$ is asumed to be continuous. We also give an analytic formula characterizing the solution $h$.

math.CA↗
arXiv2014

Dirichlet problem associated with Dunkl Laplacian on $W$-invariant open sets

Combining probabilistic and analytic tools from potential theory, we investigate Dirichlet problems associated with the Dunkl Laplacian $Δ_k$. We establish, under some conditions on the open set $D\subset\R^d$, the existence of a unique continuous function $h$ in the closure of $D$, twice differentiable in $D$, such that $$ Δ_kh=0 \quad\textrm{in}\;D\quad\textrm{and}\quad h=f\quad\textrm{on}\; \partial D. $$ We also give a probabilistic formula characterizing the solution $h$. The function $f$ is assumed to be continuous on the Euclidean boundary $\partial D$ of $D$.

math.PR↗