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Explicit formulas of strict BV relaxed energies for polyconvex functionals with linear growth

This paper derives explicit formulas for the strict BV-relaxed energies of polyconvex functionals with linear growth by extending Acerbi and Dal Maso's continuity results to the BV setting and applying Reshetnyak's continuity theorem, while also presenting partial findings on the relaxation of the vortex map under $L^1$ convergence.

Original authors: Domanico Mucci, Riccardo Scala

Published 2026-08-05

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Domanico Mucci, Riccardo Scala

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to draw a perfect map of a mountain range on a piece of paper. In the world of mathematics, this is similar to studying "functionals," which are fancy formulas that measure the "energy" or "cost" of a shape or a surface. Usually, if you have a smooth, nice shape, calculating this energy is straightforward. But what happens when the shape gets messy? What if it has sharp corners, sudden jumps, or even holes? This is the realm of "relaxed energies." Think of it like trying to smooth out a crumpled piece of foil. If you just squish it with your hand (a standard mathematical approach), you might miss the tiny, hidden wrinkles that actually hold the most energy. Mathematicians have been trying to figure out exactly how to calculate the true energy of these messy shapes, especially when the rules of the game change and the energy grows in a "linear" way (like a straight line) rather than exploding wildly. This is crucial for understanding everything from how rubber bands stretch to how soap films form, because nature loves to find the path of least resistance, even if that path looks a bit jagged.

Now, enter the story of this paper, where two mathematicians, Domenico Mucci and Riccardo Scala, act as detectives solving a very specific puzzle about these jagged shapes. They are looking at a special type of formula called "polyconvex," which is a rule that ensures the energy behaves nicely even when things get complicated. The big mystery they tackle is: if we have a messy shape (mathematicians call this a function in "BV," or Bounded Variation), can we write down a simple, exact formula for its energy if we are allowed to smooth it out in a very strict, careful way?

In the past, mathematicians knew how to do this for some simple cases, but for many others, the answer was a foggy "we know it exists, but we can't write it down." The authors of this paper say, "Hold on, we can actually write it down!" However, they don't claim to have a magic wand for every single shape. Instead, they discovered that if the relaxed area (the energy of the shape) is already known for a specific case, then we can use that knowledge to find an exact recipe for a much wider class of related energies. They proved that for these specific situations, the energy is simply the sum of two things: the energy of the smooth parts you can see, plus a specific "bonus" energy that comes from the hidden, vertical parts of the shape's shadow.

They didn't just guess; they proved this using a powerful tool called Reshetnyak's continuity theorem, which is like a magic bridge that connects the smooth world to the messy world without losing any information. They applied their new formula to three specific types of tricky maps where the relaxed area was already understood:

The Circle Walkers: Maps that stay on the edge of a circle (like a compass needle spinning).
The Homogeneous Maps: Shapes that look the same no matter how much you zoom in or out, like a fractal.
The Patchwork Quilts: Maps that are smooth in different pieces but jump abruptly where the pieces meet.

For these cases, they showed that the "bonus" energy is exactly the area of a minimal surface (like a soap film) that fills in the gaps or holes created by the jumps. However, they also found a boundary to their success. They showed that if you try to be too loose with your smoothing (using a less strict method), the answer becomes messy and unpredictable, like trying to measure a shadow that keeps changing shape. In one famous example called the "vortex map" (which looks like a whirlpool), they showed that if the area is big enough, the energy gap is a fixed number (specifically $\pi$ ), but if the area is small, the answer depends on a complex 3D surface problem that is still being solved.

So, what is the bottom line? The authors have provided a clear, explicit formula for the energy of many complex shapes, but only when you are careful about how you smooth them out and when the underlying geometry of the shape is already well-understood. They have turned a vague "it's complicated" into a concrete "here is the math" for those specific, known cases, but they also admit that for some very specific, wild shapes (like the "double-eight" curve), the answer might still be hiding in a different kind of mathematical shadow. Their work doesn't solve every mystery in the universe of shapes, but it gives us a very sharp flashlight to see the ones we were previously stumbling over in the dark.

=== DRAFT ===
Technical Summary: Explicit Formulas of Strict BV Relaxed Energies for Polyconvex Functionals with Linear Growth

Problem Statement
The paper addresses the relaxation of polyconvex variational functionals defined on vector-valued functions $u: \Omega \to \mathbb{R}^N$ (where $\Omega \subseteq \mathbb{R}^n$ is a bounded open set, $n, N \ge 2$ ) with linear growth. The model case is the area functional, where the integrand depends on the minors of the gradient matrix $G \in \mathbb{R}^{N \times n}$ .

While the relaxation of such functionals with respect to $L^1$ convergence ( $\tilde{F}_{L^1}$ ) is well-studied, it is known that the resulting relaxed functional often lacks an integral representation and fails to be local (i.e., the set function $A \mapsto \tilde{F}_{L^1}(u, A)$ is not subadditive). This non-locality arises because $L^1$ convergence allows for concentration effects that generate multiple distinct limit Cartesian currents for the same sequence of approximations.

To overcome these drawbacks, the authors analyze the relaxation with respect to **strict convergence in $BV $** ($ \tilde{F}_{BV}$). In this setting, the convergence of the total variation of the derivative is required alongside $L^1$ convergence. The primary goal is to derive explicit integral representation formulas for $\tilde{F}_{BV}(u, \Omega)$ for wide classes of polyconvex integrands, specifically in cases where the relaxed area functional $\tilde{A}_{BV}(u, \Omega)$ is already known.

Methodology
The authors employ a combination of geometric measure theory, specifically the theory of Cartesian currents, and continuity results for functionals defined on measures.

Cartesian Currents and Uniqueness: The analysis relies on the representation of graphs of smooth approximations as integral currents. A key observation is that in codimension $N=2$ (i.e., maps $u: \mathbb{R}^n \to \mathbb{R}^2$ ), strict $BV$ convergence ensures the uniqueness of the limiting Cartesian current $T$ associated with a sequence of approximations. This contrasts with the $L^1$ case or higher codimensions ( $N \ge 3$ ), where multiple limit currents may exist.
Reshetnyak Continuity Theorem: The core technical tool is an extension of Reshetnyak's continuity theorem. The authors prove that if a sequence of currents $T_h$ converges weakly to $T$ and their masses converge ( $M(T_h) \to M(T)$ ), then the parametric variational integral associated with a polyconvex integrand $f$ also converges: $F_f(T_h) \to F_f(T)$ .
Approximation Strategy: To establish the explicit formula for $\tilde{F}_{BV}$ , the authors verify that for specific classes of maps, there exists a recovery sequence $\{u_h\}$ converging strictly in $BV$ such that the associated graph currents $G_{u_h}$ converge to a unique optimal Cartesian current $T$ with minimal mass. By combining the lower semicontinuity of the relaxed energy with the continuity result derived from Reshetnyak's theorem, they equate the relaxed energy to the parametric integral evaluated on this unique current.
Plateau Problem Analysis: For specific map classes (0-homogeneous maps and piecewise Lipschitz maps), the authors utilize results concerning the planar Plateau problem. They analyze the algebraic structure of the homotopy class of the boundary data to determine when the mass of the minimal surface coincides with the mass of the associated current, a condition necessary for the explicit formula to hold.

Key Contributions and Results

Main Theorem (Theorem 1.1 / Theorem 3.7): The paper establishes that if $u \in BV(\Omega; \mathbb{R}^2)$ admits a unique Cartesian current $T$ that minimizes the mass among all currents with underlying map $u$ (a condition satisfied in codimension $N=2$ under strict $BV$ convergence), then the relaxed energy for any continuous polyconvex integrand $f$ is given explicitly by:
$\tilde{F}_{BV}(u, \Omega) = F_f(T)$
where $F_f(T)$ is the parametric variational integral associated with $f$ evaluated on $T$ .
Explicit Formulas for Specific Classes: The authors apply this main result to three specific classes of maps where the structure of the optimal current $T$ is known, leveraging existing representation formulas for the relaxed area functional:
1. Sobolev maps into $S^1$ : For $u \in W^{1,1}(\Omega; S^1)$ , the relaxed energy is expressed in terms of the distributional determinant (or the $(n-2)$ -current $P(u)$ detecting singularities). The energy gap is explicitly calculated as an integral over the vertical current $P(u) \times [[D^2]]$ .
2. 0-Homogeneous maps: For maps of the form $u_\gamma(x) = \gamma(x/|x|)$ , the relaxed energy involves a planar singular Plateau problem. The authors provide sufficient conditions (property (H) on the homotopy class of $\gamma$ ) under which the energy gap is explicitly computable via the mass of a specific current related to the minimal surface spanning the boundary curve.
3. Piecewise Lipschitz maps: For maps defined on a partition of the domain with a network of interfaces, the authors derive a formula involving integrals over the jump surfaces and contributions from junction points, provided the boundary data at junctions satisfies property (H).
$L^1$ Relaxation of the Vortex Map: The paper provides partial results regarding the $L^1$ relaxation of the vortex map $u_V(x) = x/|x|$ . It is shown that for sufficiently large domains, the energy gap is determined by the mass of a vertical current supported on a disk in the target space, whereas for small domains, the gap is related to a catenoidal surface. The authors note that the non-locality of the $L^1$ relaxation persists in these cases.

Significance and Claims
The paper provides explicit integral representation formulas for strict $BV$ relaxed energies of general polyconvex functionals with linear growth, extending known results for the area functional to a broader class of integrands.

Locality: The authors emphasize that in codimension $N=2$ , the strict $BV$ topology restores locality (the relaxed functional becomes the restriction of a Borel measure), unlike the $L^1$ topology. This allows for the derivation of explicit formulas that were previously unavailable for general integrands, building upon cases where the relaxed area representation was already established.
Continuity Extension: The work extends the continuity properties of polyconvex energies observed by Acerbi and Dal Maso in the Sobolev setting to the $BV$ setting, specifically leveraging the uniqueness of the limit current in codimension 2.
Scope: The results are presented as valid for wide classes of integrands (polyconvex, linear growth) but are contingent on the existence of a unique optimal Cartesian current. The authors explicitly acknowledge that for $N \ge 3$ or for maps failing specific algebraic conditions (like the "double-eight" example), the uniqueness of the limit current may fail, and thus the explicit formula may not hold or requires further investigation.

The paper concludes by identifying open questions, particularly regarding the extension of these results to cases where the uniqueness of the minimal injection in the Plateau problem fails (e.g., the double-eight curve) and the precise characterization of the $L^1$ relaxation for the vortex map on small domains.

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