Preprint

Inserted: 1 may 2025
Last Updated: 5 may 2025

Year: 2025

Abstract:

In this paper we develop the Direct Method in the Calculus of Variations for free-discontinuity energies whose bulk and surface densities exhibit superlinear growth, respectively for large gradients and small jump amplitudes. A distinctive feature of this kind of models is that the functionals are defined on $SBV$ functions whose jump sets may have infinite measure. Establishing general lower semicontinuity and relaxation results in this setting requires new analytical techniques. In addition, we propose a variational approximation of certain superlinear energies via phase field models.

Keywords: phase-field approximation, Superlinear free-discontinuity functionals, $SBV$ functions with infinite measure of the jump set

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