Optimal Eulerian Liftings of Second-Order Functionals

Definitions and mathematical statement

Let $u\colon\Omega\subset{\mathbb{R}}^q\to{\mathbb{R}}^d$ be a map for which the second-order energy \[ E(u)=\int_\Omega H\left(\nabla^2u(x)\right)\,{\,\mathrm{d}} x \] is defined. Here $\nabla^2u$ denotes the Hessian tensor, with components $\partial_i\partial_j u^\alpha$ for $1\leq i,j\leq q$ and $1\leq\alpha\leq d$; the source does not specify whether derivatives are classical or weak.

The supplied question asks for the largest convex, lower-semicontinuous, subadditive lifting of $E$, described in the supplied background as an optimal Eulerian lifting. Interpreting "largest" in the pointwise order, its target has the schematic form \[ F_*\in\mathscr L(E),\qquad F(v)\leq F_*(v) \quad\text{for every }F\in\mathscr L(E) \text{ and every admissible lifted argument }v, \] where $\mathscr L(E)$ denotes the intended class of convex, lower-semicontinuous, subadditive liftings.

Unresolved formulation. The supplied record does not define this class: it omits the lifted domain, its topology and addition, the representation of maps as lifted objects, and the precise condition expressed by "lifts $E$." Thus $\mathscr L(E)$ is a placeholder for missing source definitions, not a fully specified admissible class. The record also gives no hypotheses on $H$, no regularity assumptions on $\Omega$, no function space for $u$, and no boundary conditions. Consequently, the displayed target records the intended extremal question but cannot determine a unique mathematical problem without further source context.

Short English statement

For an energy depending on the Hessian of a map, determine its greatest convex, lower-semicontinuous, subadditive Eulerian lifting. The supplied record leaves the lifting framework and analytic hypotheses unspecified, so this is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005791$, source problem code catalogue entry, titled "Optimal Eulerian Liftings of Second-Order Functionals." The record assigns the problem to the Oberwolfach Reports Open Problems collection and gives proposed year 2024; no individual proposer is identified.
• Applications of Optimal Transportation (2024), Oberwolfach Reports, pp. 309--388, DOI 10.4171/owr/2024/7. Supplied source citation and literature-source link, labelled "original OWR report": https://doi.org/10.4171/owr/2024/7. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Lavenant, Lifting functionals defined on maps to measure-valued maps. Supplied literature-source link: https://arxiv.org/abs/2309.02260.
• Literature assessment checked at 2026-08-22: the supplied curation labels the problem open and states that the report and subsequent primary lifting paper leave the optimal Eulerian lifting for second-order energies as an explicit open problem. This is supplied dated curation, not an independently verified literature assessment; the background requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official Oberwolfach report was checked directly and that the clean statement restores definitions and removes extraction artifacts or source-dependent wording. These are supplied verification claims, not verification performed for this draft; the supplied excerpt still lacks the definitions needed to specify the lifting problem completely.