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A compactness result in \(GSBV^p\) and applications to \(\varGamma \)-convergence for free discontinuity problems


We present a compactness result in the space \(GSBV^p\) which extends the classical statement due to Ambrosio (Arch Ration Mech 111:291–322, 1990) to problems without a priori bounds on the functions. As an application, we revisit the \(\varGamma \)-convergence results for free discontinuity functionals established recently by Cagnetti et al. [Ann Inst H Poincaré Anal Non Linéaire (to appear)]. We investigate sequences of boundary value problems and show convergence of minimum values and minimizers.

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Correspondence to Manuel Friedrich.

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Friedrich, M. A compactness result in \(GSBV^p\) and applications to \(\varGamma \)-convergence for free discontinuity problems. Calc. Var. 58, 86 (2019).

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Mathematics Subject Classification

  • 49J45
  • 49Q20
  • 70G75
  • 74Q05
  • 74R10