Finite-Order Laminates for Rank-One Convex Envelopes

Definitions and mathematical statement

Finite-attainment problem. For an energy density $W$, find readily checkable sufficient conditions under which its rank-one convex envelope $W^{\mathrm{rc}}$ is attained by a finite-order laminate; bound the required order $N$ in terms of the specified energy data and control the radius containing all matrices in the construction. The curated wording asks for a finite prescribed radius, while the original asks for a bound on the radius; the intended dependence and quantifiers need to be fixed.

In a usual matrix-space formulation, let $W:{\mathbb{R}}^{m\times n}\to{\mathbb{R}}\cup\{+\infty\}$. A function $h$ is rank-one convex if $h(\theta A+(1-\theta)B)\le\theta h(A)+(1-\theta)h(B)$ whenever $0\le\theta\le1$ and $\operatorname{rank}(A-B)\le1$, with the intended domain and extended-value conventions. The envelope is the greatest such minorant, equivalently $W^{\mathrm{rc}}(F)=\sup\{h(F):h\le W,\ h\text{ rank-one convex}\}$ when this convention is well defined. It is not the quasiconvex or polyconvex envelope.

A finite laminate at barycenter $F$ is a finitely supported probability measure $\nu=\sum_{i=1}^L\theta_i\delta_{F_i}$ obtainable from $\delta_F$ by finitely many binary replacements of an atom at $C$ by atoms at $A,B$, where $C=\theta A+(1-\theta)B$ and $\operatorname{rank}(A-B)\le1$. Its barycenter satisfies $\sum_i\theta_iF_i=F$. Attainment at a target matrix means\[W^{\mathrm{rc}}(F)=\sum_{i=1}^L\theta_iW(F_i),\] not merely convergence of such averages along laminates of increasing complexity. An order bound must specify whether $N$ counts tree depth, total splitting steps, or another convention; these are not identical.

The requested quantitative result should specify the target matrices and whether $N$ is uniform over them, the admissible hypotheses on $W$, and the permitted dependence of a radius bound. If a ball $\overline B_R(0)$ is prescribed for the support, necessarily $F\in\overline B_R(0)$, since it is a convex barycenter; intermediate splitting matrices then also lie in that ball. A single finite radius cannot contain supports for every barycenter in the whole unbounded matrix space. The source does not fix matrix dimensions, norm and ball center, energy regularity or coercivity, determinant or other constraints, or the domain on which uniform attainment is sought. None of these conditions is guessed.

The supplied triage checked 2026-08-21 labels the item partially solved, citing numerical rank-one-convexification and laminate approximation methods in important finite-plasticity models while describing general exact computation as open. Numerical approximation is not finite-order exact attainment, and that broader computational assessment does not supply the sufficient conditions requested here. The dated claims require independent verification.

Short English statement

Find checkable energy hypotheses giving exact attainment of the rank-one convex envelope by a finite splitting tree, with quantitative complexity and support bounds. Approximate numerical laminates do not establish exact finite attainment. The target barycenters and order and radius conventions are missing.

Sources

• Local UnsolvedMath registry, permanent ID 30000358, code catalogue entry; Finite-Order Laminates for Rank-One Convex Envelopes, Oberwolfach Reports Open Problems.
• Mini-Workshop: Analysis and Computation of Microstructures in Finite Plasticity (2006), Oberwolfach Reports, DOI 10.4171/owr/2005/52, pp. 2979--3018. https://doi.org/10.4171/owr/2005/52. Proposed year and year tag: 2005; no proposer named.
• Supplied literature label Oberwolfach Report 52: https://doi.org/10.4171/owr/2005/52.
• Supplied literature label Numerical rank-one convexification: https://arxiv.org/abs/2405.16866.
• Literature triage checked 2026-08-21, status partially_solved; record updated 2026-09-16. The corrected_verified field reports checking the official report, not independently checked here.