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 (W₀¹L_M^*(Ω),||.||_M) for the Gâteaux functional J(f) ≡∫ _Ωv(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 (W₀¹L_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 (W₀¹L_M^*(Ω),||.||_M) for the Gâteaux functional J(f) ≡∫ _Ωv(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 (W₀¹L_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