Semicontinuous functionals

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

The weak convergence can be compared with the strong one (convergence in the norm). It is easily seen that if — m strongly in V then Un u 

This property is obvious in so far as the strongly converging sequence is weakly converging. 

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

It was proved that the inequality (1.79) is equivalent to the convexity of J. On the other hand, the convexity is equivalent to (1.78). Hence the result follows from the previous theorem. 

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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... 

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). 

Definition 2.2.9 (Upper semicontinuous function) f(x) is upper semicontinuous at jc° if either of the following equivalent conditions hold ... 

Let fi(x) be a family of lower (upper) semicontinuous functions on S. Then (i) Its least upper bound (greatest lower bound)... 

The reader may seem surprised with the appearance of +oo in this definition. However, infinite values are well defined in the theory of convex functionals [11] and they are usually introduced to deal in a simple way with domain questions. With this definition the functional J] is a convex lower semicontinuous functional on the whole space L1 fl L3. We are now ready to introduce the following key theorem which we will use to prove differentiability of FL at the set of E-V-densities ... 

We see that this is simply the Lieb functional with the two-particle interaction omitted. All the properties of the functional Fh carry directly over to Th. The reason is that all these properties were derived on the basis of the variational principle in which we only required that 7 + IV is an operator that is bounded from below. This is, however, still true if we omit the Coulomb repulsion W. We therefore conclude that Th is a convex lower semicontinuous functional which is differentiable for any density n that is ensemble v-representable for the noninteracting system and nowhere else. We refer to such densities as noninteracting E-V-densities and denote the set of all noninteracting E-V-densities by >0. Let us collect all the results for 7) in a single theorem ... 

Theorem 14. TL is a convex lower semicontinuous functional with the following properties ... 

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... 

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... 

In the fed-batch (semicontinuous) operation mode, substrates are fed into the reactor but no material is removed from the reactor. Therefore, the total volume of the material within the reactor increases as a function of time. For this reactor type the mass balance for each component of the reaction mixture is given by... 

Phosgene addition is continued until all the phenolic groups are converted to carbonate functionalities. Some hydrolysis of phosgene to sodium carbonate occurs incidentally. When the reaction is complete, the methylene chloride solution of polymer is washed first with acid to remove residual base and amine, then with water. To complete the process, the aqueous sodium chloride stream can be reclaimed in a chlor-alkali plant, ultimately regenerating phosgene. Many variations of this polycarbonate process have been patented, including use of many different types of catalysts, continuous or semicontinuous processes, methods which rely on formation of bischloroformate oligomers followed by polycondensation, etc. 

From the semicontinuity theorem it follows that for every j and h the function... 

This section presents (i) the definitions and properties of convex and concave functions, (ii) the definitions of continuity, semicontinuity and subgradients, (iii) the definitions and properties of differentiable convex and concave functions, and (iv) the definitions and properties of local and global extremum points. 

Notice though that the optimal value of the dual problem cannot equal that of the primal due to the loss of lower semicontinuity of the perturbation function v(y) aty = 0. 

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