Lower Semicontinuous Regularization for Vector-Valued Mappings

The paper is devoted to studying the lower semicontinuity of vector-valued mappings. The main object under consideration is the lower limit. We first introduce a new definition of an adequate concept of lower and upper level sets and establish some of their topological and geometrical properties. A characterization of semicontinuity for vector-valued mappings is thereafter presented. Then, we define a concept of vector lower limit, proving its lower semicontinuity, and furnishing in this way a concept of lower semicontinuous regularization for mappings taking their values in a complete lattice. The results obtained in the present work subsume the standard ones when the target space is finite dimensional. In particular, we recapture the scalar case with a new flexible proof. In addition, extensions of usual operations of lower and upper limits for vector-valued mappings are explored. The main result is finally applied to obtain a continuous D.C. decomposition of continuous D.C. mappings.