Rigid Elasticity as a Variational Limit

Definitions and mathematical statement

The supplied question asks whether rigid-elasticity models arise as variational limits of models whose elastic constants tend to infinity, with $\Gamma$-convergence suggested as one possible notion of limit. Neither the class of models nor its mathematical data are specified.

A conditional formulation of the suggested $\Gamma$-convergence target is as follows. Once a state space $X$, a topology $\tau$, finite-elasticity energy functionals $E_n\colon X\to\overline{{\mathbb{R}}}$, and a proposed rigid-elasticity energy $E_{\mathrm{rig}}\colon X\to\overline{{\mathbb{R}}}$ have been specified, ask whether \[ E_n\xrightarrow{\Gamma(\tau)}E_{\mathrm{rig}} \] as the elastic constants associated with $E_n$ tend to infinity in the intended sense. Here $\overline{{\mathbb{R}}}={\mathbb{R}}\cup\{-\infty,+\infty\}$. In a setting where sequential characterization of $\Gamma$-convergence is appropriate, this requires both \[ \forall x\in X\;\forall (x_n)\subset X,\qquad x_n\xrightarrow{\tau}x\ \Longrightarrow\ E_{\mathrm{rig}}(x)\leq\liminf_{n\to\infty}E_n(x_n), \] and \[ \forall x\in X\;\exists (x_n)\subset X,\qquad x_n\xrightarrow{\tau}x, \qquad \limsup_{n\to\infty}E_n(x_n)\leq E_{\mathrm{rig}}(x). \] This is a formulation template, not a specified conjecture for a particular model. The record leaves unresolved the spatial dimension and domain, state variables, constitutive assumptions, boundary conditions, energy scaling, topology, meaning of divergence of the elastic constants, and definition of the proposed rigid-elasticity limit. It also does not say whether the target concerns a particular family or every family in some class. Consequently, no precise model-specific assertion or universal quantifier over models can be recovered from the supplied text alone.

Short English statement

Can models with increasingly large elastic constants converge variationally to rigid-elasticity models, for example through $\Gamma$-convergence of their energies? The model class, energies, topology, and boundary conditions must be specified before this becomes a precise mathematical problem. The supplied record labels the problem partially solved; that label is dated curation, not independent verification.

Sources

• Local registry: supplied UnsolvedMath record, permanent ID $30002445$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The project identifies data/unsolvedmath/problems.json as its registry and catalog/problems.json as its supplementary catalog. No individual proposer is supplied; the proposed year is recorded as 2013.
• Supplied source citation: Mini-Workshop: Inelastic and Non-equilibrium Material Behavior: from Atomistic Structure to Macroscopic Constitutive Relations (2014), Oberwolfach Reports, pp. 3147--3188, DOI 10.4171/owr/2013/54. Original OWR report: https://doi.org/10.4171/owr/2013/54. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, labeled "Conti Dolzmann and Kreisbeck": https://doi.org/10.1137/100810320.
• Supplied literature source, labeled "finite crystal plasticity Gamma convergence": https://doi.org/10.3934/dcdss.2013.6.1.
• Literature assessment checked at 2026-08-21: the supplied curation reports that $\Gamma$-convergence from finite elastic constants to rigid-elastic single-slip models has been proved in two and three dimensions, but that no general theorem covers all models described in the record. It assigns the status "partially solved." This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; the cited claims and time-sensitive status require independent verification.
• The registry marks the statement corrected_verified, with the supplied explanation that mathematical content was preserved while notation, typography, or omitted definitions were repaired for publication. This metadata does not establish mathematical correctness or independent source verification of this draft.