Lower semicontinuity

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... 

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... 

Definition 2.2.8 (Lower semicontinnous function) f(x) is lower semicontinuous at x° if either of the following equivalent conditions hold ... 

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. 

Since we required that n is an E-V-density we have proven that FL is lower semicontinuous on the set of E-V-densities. It turns out that one can prove that Fl is lower semicontinuous on all densities in S (see Theorem 4.4 of Lieb [1]). However, the proof of this is not simple and we will therefore not try to reproduce it here. Since the result is important we present it here in the form of a theorem. 

One can prove an even stronger theorem in which we only need weak convergence of the series nk. We will, however, not need that property in the remainder of this review. The notion of lower semicontinuity is an important property for convex functionals which will allow us to make several other useful statements about other properties of FL. One particular consequence of lower semicontinuity that we will use, is that for a lower semicontinuous convex functional G B —<> 1Z the following set of points... 

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 ... 

Theorem 12 (Bishop-Phelps I). Let F B —<> 1Z be a lower semicontinuous convex functional on a real Banach space B. The functional F can take the value +oo but not everywhere. Then the continuous tangent functionals to F are B -norm dense in the set of F-bounded functionals in B. ... 

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 ... 

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