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
- 1 · Permanent link
Overview
Abstract
Let Ω ⊂ ℝᴺ, N ≥ 2, be a bounded connected domain with smooth boundary. Consider the Dirichlet heat evolution with initial temperature identically one. We prove that if its signed outward normal derivative is spatially constant on the entire boundary at one positive time, then Ω is a ball. No connectedness assumption is imposed on the boundary.
The argument uses the classical method of moving planes, with an L² comparison at the incompatible initial corner, a component reconstruction lemma and an explicit parabolic corner barrier. The result gives a negative answer in the smooth connected Euclidean class to the finite-observation nonball scenario in Cavallina–Pinamonti’s Problem 3 and Conjecture 7.4. The initial datum is essential: the statement does not apply to tailored nonuniform positive temperatures on a fixed shell.
Keywords heat equation · overdetermined problem · single-time observation · constant boundary flux · moving planes · ball rigidity
Scope of the work
Bounded connected C∞ Euclidean domains in dimension N ≥ 2; initial temperature exactly one; homogeneous Dirichlet boundary; one common signed outward flux on the entire boundary at one positive time. Boundary connectedness is not assumed. The consequence for Cavallina–Pinamonti is limited to this smooth connected class. No rough-domain extension, arbitrary-data rigidity, quantitative stability or finite-sensor reconstruction is claimed.
Reproduction notes
The five-member publication package contains the six-page PDF, complete main.tex with bibliography, README.md, CITATION.bib and MANIFEST.sha256. Compile main.tex with Tectonic or compatible LaTeX. The proof is analytic and has no computational or dataset premises.
The manuscript
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Author disclosures
AI tools assisted mathematical exploration, drafting and checking. The author is responsible for the statements and presentation. Internal AI checks are not external human peer review.
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Cite this edition
Scott Schubr. (2026). Single-time boundary heat-flux rigidity for smooth domains (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/single-time-boundary-heat-flux-rigidity-smooth-domains/v/1
BibTeX
@techreport{luminary_single_time_boundary_heat_flux_rigidity_smooth_domains_v1,
title = {Single-time boundary heat-flux rigidity for smooth domains},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/single-time-boundary-heat-flux-rigidity-smooth-domains/v/1}
}Loading discussion…