A.Addou, S. Lahrech
Sufficient conditions for elliptic problem
of optimal control in $\R^n in Orlicz Sobolev space
(Lobachevskii Journal of Mathematics, Vol.6, pp.19-32)

We consider here a problem for which we seek the local minimum in Orlicz Sobolev spaces $(W^1_0L_M^*(\Omega),\|.\|_{M})$ for the G\^ateaux functional $J(f)\equiv \dint\limits_{\Omega} v(x,u,f)dx$,where $u$ is the solution of Dirichlet problem with Laplacian operator associated to $f$ and $\|.\|_{M}$ is the Orlicz norm. Note that, under the rapid growth conditions on $v$, the (G.f) $J$ is not necesseraly Frechet differentiable in $(W^1_0L_M^*(\Omega),\|.\|_{M})$. In this note, using a recent extension of Frechet Differentiability,(see \cite{s}) ,we prove that, under the rapid growth conditons on $v$ the (G.f) is differentiable for the new notion. Thus we can give sufficient conditions for local minimum.

DVI format (66Kb), ZIP-ed DVI format (22Kb),
ZIP-ed PostScript format (66Kb)