A new space of generalised vector-valued functions of bounded variation

Technical Summary: A New Space of Generalised Vector-Valued Functions of Bounded Variation

Problem Statement and Motivation
The paper addresses the mathematical analysis of energy functionals arising in fracture mechanics, specifically those modeling cohesive crack growth. The study focuses on functionals of the form:
$$F(u) = \int_A f(x, \nabla u) \, dx + \int_A f^\infty\left(x, \frac{dD^c u}{d|D^c u|}\right) d|D^c u| + \int_{J_u} g(x, [u], \nu_u) \, d\mathcal{H}^{d-1}$$
where $A \subset \mathbb{R}^d$ is an open bounded set, $u: A \to \mathbb{R}^k$ is a displacement field, $J_u$ is the jump set (crack surface), $[u]$ is the crack opening, and $D^c u$ is the Cantor part of the derivative.

The specific challenge arises when the surface energy density $g$ satisfies linear growth conditions near the origin (e.g., $g \sim |\zeta| \wedge 1$), which is physically relevant for cohesive models but mathematically delicate. In such cases, the functional does not control the full amplitude of crack openings, and standard spaces like $SBV(A; \mathbb{R}^k)$ or even the scalar space $GBV(A)$ fail to provide compactness for minimizing sequences without adding artificial lower-order terms.

In the scalar case ($k=1$), Dal Maso and Toader [18] introduced the space $GBV^\star(A)$ to overcome this, proving that minimizing sequences can be modified to converge almost everywhere without additional coercivity terms. The primary problem addressed in this paper is the extension of this framework to the vectorial case ($k > 1$), i.e., defining and analyzing the space $GBV^\star(A; \mathbb{R}^k)$, and establishing the necessary compactness and lower semicontinuity results to apply the Direct Method of the Calculus of Variations.

Methodology and Definitions
The author, Davide Donati, defines the space $GBV^\star(A; \mathbb{R}^k)$ by extending the scalar definition component-wise. A function $u \in L^0(A; \mathbb{R}^k)$ belongs to $GBV^\star(A; \mathbb{R}^k)$ if and only if each component $u_i$ belongs to the scalar space $GBV^\star(A)$.

Key methodological steps include:

1. Characterization via Composition: The paper characterizes $GBV^\star(A; \mathbb{R}^k)$ not merely by component-wise properties but through the behavior of compositions with smooth, compactly supported functions. Specifically, $u \in GBV^\star(A; \mathbb{R}^k)$ if and only if for every Lipschitz function $\phi: \mathbb{R}^k \to \mathbb{R}^k$ with compact support, the composition $\phi \circ u$ belongs to $BV(A; \mathbb{R}^k)$, and the total variation of these compositions is uniformly controlled by the Lipschitz constant of $\phi$.
2. Truncation Analysis: The author utilizes a family of smooth radial truncation functions $\psi_R$ (which act as the identity on balls of radius $R$ and vanish outside radius $\sigma R$) to relate the generalized derivative $D^c u$ to the derivatives of the truncated functions $\psi_R \circ u$.
3. Adaptation of Scalar Arguments: The proofs for the vectorial case largely adapt the arguments from the scalar theory [18] and related works [17], utilizing slicing techniques and the chain rule for $BV$ functions to handle the vector-valued nature of the problem.

Key Contributions and Results

1. Definition and Structural Properties of $GBV^\star(A; \mathbb{R}^k)$:

   • The space is established as a vector space.
   • The paper proves that the Cantor part of the derivative, $D^c u$, satisfies the Rank-One property (analogous to Alberti's Rank-One Theorem for $BV$ functions). This is derived by showing that the Radon-Nikodym derivative of $D^c(\psi_R \circ u)$ converges to that of $D^c u$, and applying the known rank-one property for $BV$ functions to the truncated sequences.
   • The relationship between the jump set $J_u$ and the jump sets of truncated functions is clarified, showing that $J_{\psi_R \circ u} \subset J_u$ up to a negligible set.

2. Lower Semicontinuity:

   • The paper proves a lower semicontinuity result for integral functionals of the form $E_{f,g}$ defined on $GBV^\star(A; \mathbb{R}^k)$.
   • Under specific growth conditions on the bulk density $f$ (linear growth) and the surface density $g$ (linear growth near zero, bounded by $|\zeta| \wedge 1$), the functional is shown to be lower semicontinuous with respect to convergence in measure ($L^0$ topology).
   • The proof relies on a slicing argument adapted from [9], reducing the vectorial problem to a one-dimensional problem where weak* lower semicontinuity in $BV$ can be invoked.

3. Compactness and Modification of Minimizing Sequences:

   • Theorem 4.7: The paper establishes that if a sequence $(u_n) \subset GBV^\star(A; \mathbb{R}^k)$ has uniformly bounded energy and satisfies a mild coercivity condition (boundedness of $\int_A h(|u_n|) dx$ for some superlinear $h$), then there exists a subsequence converging $L^d$-almost everywhere to a limit $u \in GBV^\star(A; \mathbb{R}^k)$.
   • Theorem 4.8: Crucially, the paper demonstrates that for minimizing sequences satisfying Dirichlet boundary conditions, one can construct a modified sequence $(y_n)$ such that:
     • $y_n$ satisfies the same boundary conditions.
     • $F(y_n) \leq F(u_n) + \varepsilon_n$.
     • The modified sequence satisfies the coercivity condition required for Theorem 4.7.
   • This result implies that minimizing sequences can be "repaired" to ensure almost everywhere convergence without adding lower-order terms to the functional, a significant advantage for existence proofs in fracture mechanics.

Significance and Claims
The paper claims that the introduction of $GBV^\star(A; \mathbb{R}^k)$ and the associated compactness results provide the necessary analytical framework to study minimum problems for cohesive fracture models in the vectorial setting.

The author explicitly states that these results are intended to be used in a subsequent work [15] to address the homogenisation of functionals of cohesive type in the vectorial setting. The paper does not claim to solve specific physical problems directly but rather provides the rigorous functional analytic tools (existence of minimizers via the Direct Method) required to study such problems. The work extends the scalar theory of Dal Maso and Toader [18] to the vectorial case, ensuring that the compactness properties essential for variational methods are preserved when dealing with multi-dimensional displacement fields.