Single-time boundary heat-flux rigidity for smooth domains
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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.
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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
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.
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