Heat-Content Asymptotics for Polyhedra and Other Planar Domains

Definitions and mathematical statement

Let $D\subset{\mathbb{R}}^d$ be a planar domain when $d=2$, or a polyhedral domain when $d=3$. The admissible classes of domains remain to be specified. Once a heat evolution $u_D(t,x)$, its initial data, its boundary or transmission conditions, and its normalization have been fixed, define its total heat content by \[ H_D(t)=\int_D u_D(t,x)\,\mathrm{d}x,\qquad t>0, \] whenever this integral is finite. This is a conditional framework for expressing the question: the supplied record does not specify the heat evolution needed to determine $H_D$.

The requested problem is to determine the asymptotic behavior of $H_D(t)$ as $t\downarrow0$ for broader classes of planar domains and for polyhedral domains in ${\mathbb{R}}^3$, including exact contributions associated with angles. The phrase "broader classes" has no specified reference class in the supplied record. Neither the expansion scale nor the meaning of the angular contributions is specified; in three dimensions, the record does not distinguish contributions associated with dihedral angles from those associated with vertices and their solid-angle geometry.

To express the additional identification question conditionally, suppose a fixed heat model and a common asymptotic scale yield \[ H_D(t)=c_0(D)\phi_0(t)+c_1(D)\phi_1(t)+c_2(D)\phi_2(t)+o\!\left(\phi_2(t)\right), \qquad t\downarrow0, \] where the scale functions are nonzero for sufficiently small positive $t$ and satisfy $\phi_{j+1}(t)/\phi_j(t)\to0$ for $j=0,1$. Determine what information about a planar domain is determined by its first three asymptotic terms, equivalently by the coefficient triple $(c_0(D),c_1(D),c_2(D))$ when the scale is fixed. This display introduces notation only; it does not assert that such an expansion exists or prescribe its scale. The intended equivalence relation between domains, and whether "first three" means the first three prescribed orders or the first three nonzero terms, are not supplied.

The retained original question is: "What is the precise contribution from the angles?" Although the record says that a report passage supplies its context, that passage is absent from the supplied material. Consequently, the domain assumptions and heat model cannot be recovered here, and this draft remains an explicitly incomplete formulation requiring mathematical review.

Short English statement

Determine small-time heat-content expansions for the intended planar domains and three-dimensional polyhedra, including exact angular contributions, and determine how much the first three terms distinguish planar domains. The supplied record leaves the heat model, admissible domains, expansion scale, and intended notion of distinction unresolved.

Sources

• Local UnsolvedMath registry, supplied source record: permanent ID $30005599$; exact source problem code catalogue entry; title: Heat-Content Asymptotics for Polyhedra and Other Planar Domains. Collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the recorded proposed year is 2023.
• Supplied report citation: Geometric Spectral Theory (2024), Oberwolfach Reports, DOI 10.4171/owr/2023/36, pp. 2017--2106. Supplied source URL and literature-source link, labeled "Oberwolfach Report 36": https://doi.org/10.4171/owr/2023/36. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The report itself has not been independently consulted for this draft.
• Supplied literature assessment, checked on 2026-08-22: the precise angular contribution is described as unresolved in the report, and the checked literature is said not to supply a general answer. This is supplied dated curation, not an independently verified literature assessment; the source explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the problem "open" and the statement corrected_verified, and reports that the statement was reconstructed and checked against the official Oberwolfach report. These are supplied status and verification claims, not mathematical or bibliographic verification performed for this draft.