Single-time boundary heat-flux rigidity for smooth domains
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Abstract
Let Ω ⊂ R^N, 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. The same flux value is imposed on all boundary components. The proof uses moving planes, an L^2 initial-time comparison, and a parabolic corner barrier. We also establish local quantitative stability near a ball: after fixing volume and centroid, the full boundary flux defect in C^(m−1,α) controls the radial deformation in C^(m,α), for m ≥ 3 and 0 < α < 1, uniformly on compact positive-time intervals. The key input is positivity and order-one growth of the linearized spherical multipliers above the translation modes. By contrast, observation on a moving proper boundary cap admits an infinite-dimensional family of 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
- 2026-10-23
- Reader reuse
- No additional permission granted
Scope
For bounded connected smooth Euclidean domains in dimension at least two and initial temperature one, spatially constant whole-boundary signed Dirichlet heat flux at one positive time forces a ball. Near-ball strong-norm stability and moving-cap nonuniqueness hold under the stated normalization and regularity conditions.
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.
Reproduction
The 13-page manuscript, main.tex and stability.tex are supplied with build.py, citation metadata, README.md and the member checksum manifest in this 7-member package. README.md gives Tectonic commands. The proofs are analytic and use no external datasets or numerical certificates.