# Sufficient conditions for elliptic problem of optimal control in ^{n} in Orlicz Sobolev space

## (*Lobachevskii Journal of Mathematics, Volume VI*)

We consider here a problem for which we seek the local
minimum in Orlicz Sobolev spaces (W_{0}^{1}L_{M}^{*}(Ω),||.||_{M})
for the Gâteaux functional J(f)
≡∫ _{
Ω}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_{0}^{1}L_{M}^{*}(Ω),||.||_{M}).
In this note, using a recent extension of Frechet
Differentiability,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
We consider here a problem for which we seek the local
minimum in Orlicz Sobolev spaces (W_{0}^{1}L_{M}^{*}(Ω),||.||_{M})
for the Gâteaux functional J(f)
≡∫ _{
Ω}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_{0}^{1}L_{M}^{*}(Ω),||.||_{M}).
In this note, using a recent extension of Frechet
Differentiability,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