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*
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.

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