Variational Eddy Currents with Degenerate Conductivity

Definitions and mathematical statement

Let \[ \mathcal S=\{\sigma\in L^\infty({\mathbb{R}}^3;{\mathbb{R}}):\sigma(x)\geq 0\text{ for almost every }x\in{\mathbb{R}}^3\}. \] Here $L^\infty$ denotes the space of essentially bounded measurable functions, identified up to equality almost everywhere. Conductivities in $\mathcal S$ may vanish; no strictly positive lower bound is assumed.

Source question. For every $\sigma\in\mathcal S$, can the mapping \[ A\longmapsto\phi_A \] be defined in the variational formulation of the eddy-current equation?

Unresolved formulation. The supplied record does not specify the eddy-current equation or its variational identity, the admissible space for $A$, the defining conditions or function space for $\phi_A$, or any normalization, boundary conditions, or conditions at infinity. It also does not specify whether $\phi_A$ is an individual function or an equivalence class. Consequently, the question above preserves the supplied target, but a precise existence or well-definedness statement requires this missing context from the cited report. No particular variational formulation or uniqueness convention is imposed here.

Short English statement

The question asks whether the assignment $A\mapsto\phi_A$ in an eddy-current variational formulation extends to every essentially bounded, nonnegative conductivity on ${\mathbb{R}}^3$, including conductivities that vanish. The supplied record omits the formulation and spaces needed to specify this assignment precisely.

Sources

• Local UnsolvedMath registry, permanent ID $30001994$, source problem code catalogue entry, title: Variational Eddy Currents with Degenerate Conductivity. Collection: Oberwolfach Reports Open Problems. The record gives the proposed year as 2012 and names no individual proposer.
• Inverse Problems for Partial Differential Equations (2012), Oberwolfach Reports, Report 11, pp. 611--659, DOI: 10.4171/owr/2012/11. Supplied source citation and literature-source link: https://doi.org/10.4171/owr/2012/11.
• Supplied literature assessment, checked 2026-08-21: the record labels the problem open and states that the report explicitly leaves extension to arbitrary nonnegative bounded conductivity open, with no general well-posed definition of the map found by that date. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This verification is reported by the supplied record and has not been independently repeated here.