Semicontinuous

A functional J V Ris called weakly lower semicontinuous at the point u if the condition Un u weakly in V implies... 

Now let us prove two theorems containing sufficient conditions of weak lower semicontinuity of the functionals. 

We formulate one more theorem concerning the weak lower semicontinuity. 

Theorem 1.11. Let V be a reflexive Banach space, and K c V be a closed convex set. Assume that J V R is a coercive and weakly lower semicontinuous functional. Then the problem... 

The inequality used here follows from the weak lower semicontinuity of the functional J. Thus, the element u is found such that... 

We first show that the functional J is weakly lower semicontinuous. The function H(u, uq) is linear and continuous over uq for each fixed u G V. Now let Un —t u weakly. Then... 

Let K cV he a. convex closed subset of a reflexive Banach space V, I he a duality mapping, and P be a projection operator of V onto K. We are in a position to give a definition of a penalty operator. An operator (5 V V is called a penalty operator connected with the set K if the following conditions are fulfilled. Firstly, / is a monotonous bounded semicontinuous operator. Secondly, a kernel of / coincides with K, i.e. 

A boundedness and semicontinuity of the operator / follow from the same properties of I and P. The verification of the equivalence of conditions... 

Note that the functional H is convex and continuous, and consequently, it is weakly lower semicontinuous. 

We first note that the coercivity and weak lower semicontinuity of the functional n imply that the problem (2.248) has a (unique) solution The coercivity is provided by the following two inequalities,... 

In accordance with (3.53) the functional II/(x) + Ilg( ) is coercive and weakly lower semicontinuous on the space H, consequently, the problem (3.48) (or the problem (3.54)) has a solution. The solution is unique. Note that the equilibrium equations... 

Because g is nonnegative, one can see that ff is a coercive, strongly convex and lower semicontinuous functional. Therefore, there exists a unique solution w G H Qc) of the problem (3.204) or (3.205) (see Section 1.2). 

At the beginning we study the (5-dependence of the solution and next we consider the problem of finding extreme crack shapes. First, let us note that the problem (4.168) has a solution owing to the coercivity and the weak lower semicontinuity of II5 on the space The solution is unique for... 

Moreover, B is monotone, bounded and semicontinuous. Hence, the solvability of the equation (5.157) or, equivalently, of the problem (5.154)-(5.156) follows from Theorem 1.14. We should recall that the parameters s, 5 are fixed at this point. 

Ma.nufa.cture. The batch nitration processes for nitrocellulose have included the pot process, the centrifugal process, the Thompson displacement process, and the mechanical dipper process. Semicontinuous nitration processes are also widely used for military and industrial grades. 

The two procedures primarily used for continuous nitration are the semicontinuous method developed by Bofors-Nobel Chematur of Sweden and the continuous method of Hercules Powder Co. in the United States. The latter process, which uses a multiple cascade system for nitration and a continuous wringing operation, increases safety, reduces the personnel involved, provides a substantial reduction in pollutants, and increases the uniformity of the product. The cellulose is automatically and continuously fed into the first of a series of pots at a controlled rate. It falls into the slurry of acid and nitrocellulose and is submerged immediately by a turbine-type agitator. The acid is deflvered to the pots from tanks at a rate controlled by appropriate instmmentation based on the desired acid to cellulose ratio. The slurry flows successively by gravity from the first to the last of the nitration vessels through under- and overflow weirs to ensure adequate retention time during nitration. The overflow from the last pot is fully nitrated cellulose. 

Although the purification techniques used with the semicontinuous system are similar to that for the batch system, much less wastewater is produced because much less acid is retained on the nitrocellulose that is discharged from the centrifuge (135—137). 

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