Chapter 11: Relaxation in Sobolev, BV , and Young measures spaces
Society for Industrial and Applied Mathematics eBooks · 2014
11.1 ▪ Relaxation in abstract metrizable spaces This section is devoted to the description of the relaxation principle in a general metrizable space, or more generally, in a first countable topological space X. Roughly speaking, given an extended real-valued function F : X → R ∪ {+∞}, we wish to apply the direct method in the calculus of variations to the lower semicontinuous envelope cl(F) of the function F so that infX F = minX cl(F). Such a procedure is very important in various applications and leads to the concept of generalized solutions for the optimization problem infX F. We begin by giving some complements on the sequential version, denoted by , of the general notion of lower semicontinuous envelope cl(F) introduced in Chapter 3.