Research Article · Preprint Mathematics
Single-time boundary heat-flux rigidity for smooth domains
- Primary subject
- Mathematics
- Subject subfields
- Differential Equations
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
Abstract
For a bounded connected smooth Euclidean domain in dimension at least two, consider Dirichlet heat evolution with initial temperature identically one. If the signed outward normal derivative is spatially constant on the entire boundary at one positive time, the domain is a ball; connected boundary is not assumed. The proof uses moving planes, an L2 initial-corner comparison, component reconstruction and a parabolic corner barrier. We also give a local near-ball quantitative stability estimate in strong boundary norms, with volume and centroid normalization, at times in a fixed compact positive interval. In contrast, one fixed time observed on a moving proper boundary cap admits an infinite-dimensional family of nearby normalized nonballs with exactly the ball flux on that cap. Global stability and reconstruction from a fixed physical patch are not asserted.
Keywords heat equation · overdetermined problem · single-time observation · constant boundary flux · moving planes · ball rigidity
Scope of the work
Local full-boundary C^{m-1,alpha} flux-defect control of C^{m,alpha} near-ball shape, fixed volume/centroid, m>=3, compact positive time interval. Infinite-dimensional local family of constrained nonballs matching ball flux at one fixed time on a moving proper boundary cap.
Reproduction notes
The exact publication package includes the 13-page manuscript, editable LaTeX sources, reader/build instructions, citation metadata, public checksums and any necessary mathematical or computational supplements. Keep its 7 members together and follow the included README for dependencies, build commands and any finite verification programs. Published finite checks support the stated arguments; they do not replace universal analytic proofs or establish external human refereeing.
The manuscript
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Author disclosures
AI-assisted tools supported research exploration, literature review, manuscript preparation and internal verification. The author retains responsibility for the work. Internal AI-assisted and computational checks are distinct from external human peer review.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Single-time boundary heat-flux rigidity for smooth domains (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/single-time-boundary-heat-flux-rigidity-smooth-domains/v/2
BibTeX
@techreport{luminary_single_time_boundary_heat_flux_rigidity_smooth_domains_v2,
title = {Single-time boundary heat-flux rigidity for smooth domains},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/single-time-boundary-heat-flux-rigidity-smooth-domains/v/2}
}Loading discussion…