arXiv subjects
Publications and source records attributed to Mouna Kraiem.
In this paper, we consider the analogous of the obtacle problem in $H_0^1(Ω)$, on the space $W^{1,p}_0(Ω)$. We prove an existence and uniqueness of the result. In a second time, we define the optimal control problem associated. The results, here enclosed, generalize the one obtained by D.R. Adams, S. Lenhard
In this paper we present an approximation result concerning the first eigenvalue of the 1-Laplacian operator. More precisely, for $Ω$ a bounded regular open domain, we consider a minimisation of the functional ${\ds \int_Ω}|\nabla u|+n({\ds \int_Ω} |u|-1)^2 $ over the space $W_0^{1,1}(Ω)$. For $n$ large enough, the infimum is achieved in some sense on $BV(Ω)$, and letting $n$ go to infinity this provides an approximation of the first eigenfunction for the first eigenvalue, since the term $n({\ds \int_Ω} |u|^2-1)^2$ "tends" to the constraint $\|u\|_1=1$.
Let $Ω$ be a smooth bounded domain in $\R^N, N>1$ and let $n\in \N^*$. We are concerned here with the existence of nonnegative solutions $u\_n$ in $BV(Ω)$, to the problem $$(P\_n) \begin{cases} -{\rm div} σ+2n (\int\_ Ωu -1) {\rm sign}^+ (u)=0 \quad \text{in} Ω, σ\cdot \nabla u= |\nabla u| \quad \text{in} Ω, u \text{\rm is not identically zero}, -σ\cdot \overrightarrow {n} u=u \quad \text{on} \partialΩ, \end{cases}$$ where $\overrightarrow {n}$ denotes the unit outer normal to $\partialΩ$, and ${\rm sign}^+(u)$ denotes some $L^{\infty}(Ω)$ function defined as: $${\rm sign}^+ (u). u =u^+, 0 \leq {\rm sign}^+(u) \leq 1.$$ Moreover, we prove the tight convergence of $u\_n$ towards one of the first eingenfunctions for the first $1-$Laplacian Operator $-Δ\_1$ on $Ω$ when $n$ goes to $+\infty$.