MathDB

Bouchitté’s light-structures conjecture

For every spatial dimension n≥2n\ge 2 and every admissible elasticity tensor (isotropic or anisotropic) defining a Hooke-law energy, consider the minimum-compliance shape-optimization problem over shapes of volume ε\varepsilon. As ε↓0\varepsilon\downarrow 0, the corresponding compliance functionals are conjectured to converge in the appropriate Γ\Gamma-sense to the light-structures compliance functional on generalized shapes, represented by probability measures and expressed through Bouchitté's relaxed integrand. In particular, the limiting problem should describe optimal lower-dimensional structures, including Michell-type trusses, and possess the conjectured decomposition properties. The supplied sources do not state the explicit formula for the relaxed integrand or the precise topology and normalization of the Γ\Gamma-limit.

References

Primary source

arXiv

Additional references

calculus-of-variationselasticityshape-optimization
Posted

Progress summary

Refreshed
Claimed solved

A September 2026 preprint claims to remove the conjecture’s dimension and material-law restrictions, but the result has not been refereed.

Bouchitté’s conjecture concerns the vanishing-mass relaxation limit for optimal light structures, including the predicted Γ\Gamma-limit and its decomposition properties.

Known results

  • A preprint submitted February 19, 2021, revised January 5, 2022, claimed the vanishing-mass limit under a hard mass constraint for isotropic quadratic energies in dimensions two and three, identifying the conjectured Bouchitté integrand and settling the corresponding optimal-light-structures problem.

September 2026 general-elasticity extension

A September 2026 arXiv preprint, Shape optimization of light structures with general Hooke’s laws, claims to extend the relaxation result to general elasticity tensors and arbitrary dimensions, thereby removing the earlier restrictions. This is an unrefereed claim.

Current status (as of September 2026): The earlier restricted result and a new claimed all-dimensional, general-elasticity extension are publicly available, but the full conjecture is not independently verified.

Sources

Solutions 0

No solutions have been posted yet.