Single-time boundary heat-flux rigidity for smooth domains
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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, a smooth domain with possibly disconnected boundary is a ball. 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.
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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
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.
Disclosures
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