This chapter introduces the space of piecewise-Sobolev functions, which is the one-dimensional version of the De Giorgi-Ambrosio SBV space. Sample applications to Signal Reconstruction (the Mumford-Shah functional) and Fracture Mechanics (Griffith fracture energy) are given. The Ambrosio compactness theorem is proved, and the necessary conditions are given for free-discontinuity energies. Existence results are derived.
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