Geodesic
Sufficient conditions for elliptic problem of optimal control in~$\mathbb R^N$ in Orlicz Sobolev space
Lobachevskii journal of mathematics, Tome 6 (2000), pp. 19-32

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We consider here a problem for which we seek the local minimum in Orlicz Sobolev spaces $(W_1^0 L_M^*(\Omega),\|.\|_M)$ for the Gâteaux functional $J(f)\equiv\displaystyle\int_\Omega v(x,u,f)\,dx$, 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 [2]) ,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. 
@article{LJM_2000_6_a1,
     author = {A. Addou and S. Lahrech},
     title = {Sufficient conditions for elliptic problem of optimal control in~$\mathbb R^N$ in {Orlicz} {Sobolev} space},
     journal = {Lobachevskii journal of mathematics},
     pages = {19--32},
     publisher = {mathdoc},
     volume = {6},
     year = {2000},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/LJM_2000_6_a1/}
}
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A. Addou; S. Lahrech. Sufficient conditions for elliptic problem of optimal control in~$\mathbb R^N$ in Orlicz Sobolev space. Lobachevskii journal of mathematics, Tome 6 (2000), pp. 19-32. http://geodesic.mathdoc.fr/item/LJM_2000_6_a1/