Geodesic
Sufficient conditions for elliptic problem of optimal control in~$\mathbb R^2$
Lobachevskii journal of mathematics, Tome 6 (2000), pp. 3-17

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It is known that when we look for sufficient conditions of local extremum for Gâteaux functionals (G.f) associated to Dirichlet problem of second order in $\mathbb R^2$, the (G.f) is not necesseraly Frechet differentiable. In this note, using a recent extension of Frechet Differentiability, (see [5]), we obtain that the (G.f) is differentiable with respect to the new notion. Thus we can give sufficient conditions for obtaining local minimum. 
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     author = {A. Addou and S. Lahrech},
     title = {Sufficient conditions for elliptic problem of optimal control in~$\mathbb R^2$},
     journal = {Lobachevskii journal of mathematics},
     pages = {3--17},
     publisher = {mathdoc},
     volume = {6},
     year = {2000},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/LJM_2000_6_a0/}
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A. Addou; S. Lahrech. Sufficient conditions for elliptic problem of optimal control in~$\mathbb R^2$. Lobachevskii journal of mathematics, Tome 6 (2000), pp. 3-17. http://geodesic.mathdoc.fr/item/LJM_2000_6_a0/