Characterize O_C via level-set properties

Determine whether the class O_C (domains satisfying the geometric normal property with respect to a fixed convex set C) can be completely characterized by geometric properties of the level sets associated to these domains, such as those arising from solutions to the Dirichlet and biharmonic problems studied in the paper.

Background

A central theme of the paper is that, for domains Ω in the C-GNP class, the level sets of solutions to elliptic problems (e.g., −Δu=f in Ω with u=0 on ∂Ω and f≥0 supported in C, and the coupled biharmonic system) inherit a radial structure and star-shapedness relative to C. The authors establish regularity, curvature formulas, and stability of these level sets under domain convergence.

Given these strong forward implications (from Ω∈O_C to properties of its level sets), a natural reverse problem is whether such level-set properties constitute a complete geometric characterization of the class O_C itself.