Semicontinuity and Supremal Representation in the Calculus of Variations

Semicontinuity and relaxation of L ∞-functionals

“…The proof of this property is the main extension of Theorem 2.4 to the results in [6]. In the general case, under the assumptions of Theorem 2.4, it is not possible to deduce that Q f D f lc W in fact, in the Example 8.1 of [15], the author exhibits a w* l.s.c. ; u; / is continuous for all .u; / 2 R R N , it is easy to obtain the inequality Q f .x; ; / f .x; ; / for a.e.…”

“…functional F could be represented through a level convex function. This equivalence is not a trivial consequence of the result in [15]. functional always admits a level convex supremand turns out to be interesting and useful for applications.…”

“…In a previous paper (see [15]) we have considered the case f D f .x; / and we have given a positive answer to this question by showing that a supremal functional of the form where A is the class of open subsets of and, without a continuity assumption on f . functional F could be represented through a level convex function.…”

“…Now, since a supremal functional does not admit a unique representation, the question whether a weakly* l.s.c. The strategy we will follow is to consider the distances and to apply the results established in [15] about the relationship between the functional F and the convex functionals R.u/ D sup x;y2; x¤y ju.x/ u.y/j d u 0 .x; y/ referred to as difference quotients. In a previous paper (see [15]) we have considered the case f D f .x; / and we have given a positive answer to this question by showing that a supremal functional of the form where A is the class of open subsets of and, without a continuity assumption on f .…”

“…functional always admits a level convex supremand turns out to be interesting and useful for applications. This metric approach was introduced in [12] in the case of the 1-homogeneous supremal functionals and applied, in the general case, in [7] in order to characterize the absolute minimizers of a functional of the form (1.2) and in [15] in order to give equivalent conditions to the weak* lower semicontinuity. ; u; /; we show that F .…”