Chapter 10: Spaces BV and SBV
Hédy Attouch, Giuseppe Buttazzo, Gérard Michaille · Society for Industrial and Applied Mathematics eBooks · 2014
The modelization of a large number of problems in physics, mechanics, or image processing requires the introduction of new functional spaces permitting discontinuities of the solution. In phase transitions, image segmentation, plasticity theory, the study of cracks and fissures, the study of the wake in fluid dynamics, and so forth, the solution of the problem presents discontinuities along one-codimensional manifolds. Its first distributional derivatives are now measures which may charge zero Lebesgue measure sets, and the solution of these problems cannot be found in classical Sobolev spaces. Thus, the classical theory of Sobolev spaces must be completed by the new spaces BV(Ω) and SBV(Ω).