Approximation of free-discontinuity problems. (English) Zbl 0909.49001

Braides, Andrea

Lecture Notes in Mathematics. 1694. Berlin: Springer. xi, 149 p. (1998).

According to the De Giorgi terminology the term “free discontinuity problems” concerns the class of all those problems in calculus of variations where the unknown is a pair \((u,K)\) with \(K\) varying in a family of closed hypersurfaces in a fixed open set \({\Omega} \in {\mathbb R}^n\) and \(u: {\Omega} \setminus K \to {\mathbb R}^m\) belongs to a class of sufficiently smooth functions. Such problems, in general, take the form: \(\min [E_v(u,K) + E_s(u,K)+\) “lower order terms”], where \(E_v\) and \(E_s\) denote the volume and the surface energies, respectively. In this context the book in review encompasses the large class of problems of multi-phase systems, fracture mechanics, computer vision,…(e.g., signal and image reconstruction, fractered hyperelastic media, drops of liquid cristals, prescribed curvature problems, and so on – all of them are formulated in the Introduction). After a brief and selfcontained introduction to BV spaces in Chapter 1 the author presents, in Chapter 2, the theory of SBV and GSBV spaces of special and generalized special functions of bounded variation, respectively, giving the compactness and lower semicontinuity results. Chapter 3 contains the necessary definitions and properties of the \({\Gamma}\) convergence theory which is then used in various approximation procedures of free discontinuity problems. The significant 1-dimensional case is considered still in Chapter 3, while the passage to higher dimensions is given in Chapter 4, where the most technical results (slicing and density techniques) are proved. Chapter 5 is devoted to non-local approximations. Some numerical results concerning the approximation of the Mumford-Shah functional by elliptic functionals are given in the Appendix.

Reviewer: Z.Denkowski (Kraków)

MSC:

49-02	Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
49J45	Methods involving semicontinuity and convergence; relaxation
74R99	Fracture and damage

Keywords:

BV; SBV; approximations of free discontinuity problems; nonlocal; elliptic; \(\Gamma\)-convergence; slicing method; Mumford-Shah functional; GSBV; spaces of special and generalized special functions of bounded variation; density techniques