Many problems in physics involve free discontinuities — surfaces where a quantity jumps. A material fractures along a surface. A phase boundary separates two states. An image has edges. Modeling these mathematically requires functionals that have both a bulk term (penalizing gradients, enforcing smoothness) and a surface term (penalizing discontinuities, keeping cracks short). These are free-discontinuity problems.
Brusca et al. (arXiv: 2603.24192) prove that a wide class of discrete convolution-type energies Gamma-converge to free-discontinuity functionals. Gamma-convergence is the right notion of convergence for variational problems: it guarantees that minimizers of the discrete problems converge to minimizers of the continuous problem. The discrete energies are nonlocal — each point interacts with all others through a convolution kernel — and the continuum limit produces a free-discontinuity energy on a space of generalized functions of bounded variation.
The authors characterize both the bulk and surface energy densities through minimization problems on small cubes applied to the approximating energies. The bulk density comes from how the convolution penalizes smooth gradients. The surface density comes from how it penalizes jumps.
The through-claim: continuous fracture mechanics is a limit of discrete interactions. The free-discontinuity functional — with its separate bulk and surface terms, its special treatment of cracks and interfaces — is not an ad hoc model. It emerges inevitably from the Gamma-limit of convolution energies that make no distinction between bulk and surface. The two-term structure (smoothness plus discontinuity) is not imposed — it's derived. The discrete model sees only pairwise interactions; the continuum limit discovers that some interactions want smoothness and others want sharpness, and separates them accordingly.
Brusca, Donati, Scalabrino, Trifone & Voglino, 2603.24192. Variational analysis / Gamma-convergence / free-discontinuity problems / convolution functionals / fracture mechanics.