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In previous papers [3-5] we have given existence theorems for solutions in Sobolev spaces of multidimensional Lagrange problems of optimization in a fixed domain, bounded or unbounded. In the present paper we shall apply the previous results to the case where the partial differential equations are written in canonic form. The canonic form which we take into consideration is the one proposed by RASHEVSKY in his book [9, pp. 323 324]. THOMAS & TITT [10] consider essentially the same form though this form is not written explicitly. ILaSHEVSKY has pointed out the generality of the proposed form. In [6-8] LURm discusses necessary conditions for Lagrange problems of optimization with partial differential equations in canonic form. In particular, in [7, 8] LumE discusses necessary conditions for an optimal solution in a two dimensional problem of magnetohydrodynamic channel flow, though with no previous existence analysis. In the present paper we shall show that our existence theorems yield the existence of an optimal weak solution for LURIE'S problem of magnetohydrodynamics. This optimal solution, of which we prove the existence, is weaker, however, than those considered by LURIE, namely it is a weak solution in the sense of Gamkrelidze with state functions belonging to a Sobolev space W21.

Multidimensional Lagrange problems of optimization in a fixed domain and an application to a problem of magnetohydrodynamics