Systems of Partial Differential Equations and Their Characteristic Surfaces

Thomas, T. Y.; Titt, Edwin W.

doi:10.2307/1968340

Cited by 5 publications

(1 citation statement)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Also, we shall restrict I2 by imposing boundary conditions. We shall impose only the condition (10), or…”

Section: A Weak Form Of Lupoe's Optimization Problemmentioning

confidence: 99%

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

Cesari

1968

Arch. Rational Mech. Anal.

View full text Add to dashboard Cite

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. [1-5] that analogous concepts of upper semicontinuity, more topological in character, are needed. We shall denote these properties as properties (U) and (Q). First, given ~>0 and a point (Xo, Yo, Zo)eA let us denote by U(xo, Yo, Zo; ~) the set U(xo, Yo, Zo; ~)=U U(x, y, z), where U ranges over all (x, y, z)eN~(xo, Yo, Zo). We shall say that U(x, y, z) satisfies property (U) at a point (Xo, Yo, Zo)e A provided U(xo, Yo,Zo)= ~ cl U(xo, Yo, Zo; 6), ~>0 that is, U(xo, Yo, Zo) = (] cl d U(x, y, z). (x, y, z) ~ N6 (xo, yo, zo) We shall say that U(x, y, z) satisfies property (U) in A if U(x, y, z) satisfies property (U) at every point (Xo, Yo, Zo)eA. A set U(x, y, z) satisfying property (U) is necessarily closed as the intersection of closed sets. Below we shall also consider 6*

show abstract

“…Also, we shall restrict I2 by imposing boundary conditions. We shall impose only the condition (10), or…”

Section: A Weak Form Of Lupoe's Optimization Problemmentioning

confidence: 99%