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We present a compactness result in the space $GSBV^p$ which extends theclassical statement due to Ambrosio to problems without a priori bounds on thedeformations. As an application, we revisit the $\Gamma$-convergence resultsfor free discontinuity functionals established recently by Cagnetti, Dal Maso,Scardia, and Zeppieri. We investigate sequences of boundary value problems andshow convergence of minimum values and minimizers.

A compactness result in $GSBV^p$ and applications to $\Gamma$-convergence for free discontinuity problems