Homogenisation of vectorial free-discontinuity functionals with cohesive type surface terms

Technical Summary: Homogenisation of Vectorial Free-Discontinuity Functionals with Cohesive Type Surface Terms

Problem Statement
This paper addresses the asymptotic behavior of sequences of free-discontinuity functionals in the vector-valued setting ($k \geq 1$). Specifically, it investigates the $\Gamma$-convergence of functionals defined on the space $GBV^*(A; \mathbb{R}^k)$, which generalizes the space of functions of bounded variation ($BV$) to accommodate functions with unbounded values but controlled growth. The functionals under study take the form:
$$E(u, A) = \int_A f(x, \nabla u) \, dx + \int_A f^\infty\left(x, \frac{dD^c u}{d|D^c u|}\right) \, d|D^c u| + \int_{J_u \cap A} g(x, [u], \nu_u) \, d\mathcal{H}^{d-1}$$
where $f$ and $g$ satisfy linear growth conditions typical of cohesive fracture models. Unlike previous works that often assumed superlinear growth (allowing for compactness in $SBV$) or scalar settings, this work deals with the vectorial case where the surface term $g$ satisfies a "cohesive" growth condition: $c(|\zeta| \wedge 1) \leq g(x, \zeta, \nu) \leq C(|\zeta| \wedge 1)$. This condition prevents the control of the total variation of the gradient along minimizing sequences, necessitating the use of the larger space $GBV^*$.

Methodology
The authors extend the techniques developed in their previous scalar works [20, 21] to the vectorial setting, overcoming significant technical hurdles related to truncation.

1. Function Space and Truncation: The study is conducted within $GBV^*(A; \mathbb{R}^k)$. A central difficulty in the vectorial case is that standard component-wise truncations $u^{(m)} = (u \wedge m) \vee (-m)$ do not yield the necessary estimates for the functionals. Instead, the authors employ smooth truncations of the form $\psi_m^i \circ u$, where $\psi_m^i$ are specific smooth cut-off functions. Crucially, they cannot estimate the functional for every $i \in \{1, \dots, m\}$ simultaneously; they must select a suitable index $i$ depending on the sequence and the function to obtain valid bounds. This requires a modification of the abstract framework used in the scalar case.

2. Abstract Framework ($\mathcal{E}$): The authors define an abstract class of functionals $\mathcal{E}$ on $L^0(\mathbb{R}^d; \mathbb{R}^k) \times \mathcal{B}(\mathbb{R}^d)$. They prove a compactness theorem (Theorem 4.1) showing that any sequence in $\mathcal{E}$ admits a $\Gamma$-convergent subsequence.

3. Integral Representation:

   • Partial Representation: For functionals in the lower semicontinuous subclass $\mathcal{E}_{sc}$, they establish an integral representation for the absolutely continuous part ($E_a$) and the jump part ($E_j$) in terms of integrands $f$ and $g$ (Theorem 5.16).
   • Full Representation: To represent the Cantor part ($E_c$), they introduce smaller collections of integrands $\mathcal{F}_\alpha$ and $\mathcal{G}_\vartheta$ with specific quantitative estimates near infinity and zero. Under the assumption that the bulk integrand $f$ is independent of $x$, they prove that the $\Gamma$-limit admits a full integral representation including the Cantor term (Theorem 7.1).

4. Homogenisation:

   • Deterministic: They derive necessary and sufficient conditions for the $\Gamma$-convergence of sequences with oscillating coefficients $f(x/\varepsilon_n, \cdot)$ and $g(x/\varepsilon_n, \cdot)$. The homogenized integrands are characterized as limits of minima of auxiliary minimization problems on large cubes.
   • Stochastic: They apply these results to the stochastic setting where $f$ and $g$ are random fields satisfying stationarity and ergodicity. By utilizing the Subadditive Ergodic Theorem, they prove that the $\Gamma$-limit exists almost surely and is deterministic (independent of the realization $\omega$) if the underlying transformation group is ergodic.

Key Contributions and Results

• Extension to Vector-Valued Functions: The primary contribution is the extension of homogenisation theory for cohesive free-discontinuity functionals from the scalar case ($k=1$) to the vectorial case ($k \geq 1$).
• Technical Innovation in Truncation: The paper introduces a refined truncation technique using smooth compositions $\psi_m^i \circ u$ and proves that for any sequence of functionals, there exists a "good" index $i$ (dependent on $n, m, u$) that preserves the energy estimates. This resolves the main obstruction in generalizing scalar results to the vectorial setting.
• Integral Representation in $GBV^*$: The authors provide a complete integral representation for the $\Gamma$-limit of sequences of functionals with linear growth and cohesive surface terms, explicitly characterizing the bulk, surface, and Cantor parts.
• Stochastic Homogenisation: The paper establishes an almost sure $\Gamma$-convergence result for stochastic free-discontinuity functionals with cohesive surface terms, proving that the homogenized energy density is deterministic under ergodicity assumptions.

Significance and Claims
The authors position this work as a "preliminary step" toward understanding the asymptotic behavior of crack growth in heterogeneous cohesive materials as the grain size tends to zero. They explicitly state that their results provide a complete answer to the first part of the program for rate-independent problems: the $\Gamma$-convergence of the energy-dissipation functionals.

The paper acknowledges that the construction of a "joint recovery sequence," which is required to prove the convergence of quasi-static evolutions (the second part of the program), is not addressed here. The results are framed as a rigorous mathematical foundation for homogenisation in this specific class of non-linear, vector-valued free-discontinuity problems, building upon the scalar foundations laid in [20, 21] and the vectorial periodic/stochastic works on superlinear growth [12, 13, 14, 15]. The work relies heavily on the structural properties of $GBV^*$ and the specific growth conditions of cohesive models, distinguishing it from classical homogenisation in $SBV$ or $BD$.