Sufficient conditions for elliptic problem of optimal control in $\mathbb R^N$ in Orlicz Sobolev space

A. Addou, S. Lahrech

Mohamed First University

Abstract: 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.

Submitted by: A. V. Lapin
Received: 18.08.1999

Language: English

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