Single-time boundary heat-flux rigidity for smooth domains
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Abstract
We study recovery of a domain from the entire boundary heat flux at one positive time, for Dirichlet evolution with initial temperature one. Near a ball, after fixing volume and centroid, the flux defect in C^(m−1,α) controls the radial deformation in C^(m,α), where m≥3 and 0<α<1. Dividing the flux by its physical boundary mean gives one stability constant and one shape neighborhood valid for every observation time T≥ T₀>0. This normalization removes the exponential late-time loss of the absolute flux signal; it does not remove sensitivity to fixed absolute measurement noise. The proof combines strict positivity of the spherical heat multipliers above translations with uniform shape-differentiable convergence to the normalized first-eigenfunction flux. We also give a moving-plane proof that constant signed flux on every boundary component at one time forces a bounded connected smooth domain to be a ball, without convexity or connected boundary. In contrast, a moving proper observation cap admits infinitely many nearby normalized nonballs with exactly the ball flux at the prescribed time.
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Research details
- Subjects
- Mathematics · Differential Equations
- Keywords
- heat equation, overdetermined problem, single-time observation, constant boundary flux, moving planes, ball rigidity
- Manuscript date
- 2026-10-29
- Reader reuse
- none
Scope
Local recovery of volume- and centroid-normalized near-ball domains from the entire same-time Dirichlet heat flux with initial temperature one. Physical-mean normalization gives one strong-Hölder inverse constant and one shape neighborhood uniformly for all T≥T₀>0. The paper includes exact whole-boundary rigidity, exponential raw-flux conditioning and moving-cap nonuniqueness under their stated hypotheses.
Disclosures
Generative AI tools, including ChatGPT and Codex, assisted research exploration, proof development, literature discovery, drafting and internal verification under the author's direction. The author assumes responsibility for the mathematical content and attribution. The author received no external funding or material support and owns Luminary Oddities, which maintains the Luminary Research Archive. No other relevant competing interests are declared.
Reproduction
Seventeen-page analytic manuscript with main.tex and stability.tex, portable Tectonic build.py, reader instructions, citation metadata and a complete member-checksum manifest in a seven-member source package. The actual packaged build reproduces the supplied PDF byte for byte.