Positive heat-flow observations on spherical shells: cutoffs, optimal preparation and concentration
Loading original first page…
Abstract
We study nonnegative initial temperatures whose Dirichlet heat evolution has one common signed outward normal derivative on both boundary components of a fixed spherical shell. In every dimension at least two, the admissible observation times form an interval (0, τ) with a finite, unattained endpoint characterized by a boundary heat-kernel ratio. Finite schedules satisfy an exact interior convex-hull criterion, and a datum-independent terminal interval contains at most one observation. With unit ground-state projection, every weighted minimum-Lq preparation, 1 < q < ∞, is unique at each admissible time and is determined by a monotone real scalar moment equation. We obtain sharp cost constants throughout the fixed norm scale, including separate endpoint arguments. For every finite strictly convex power, quantitative ground-probability stability prevents scaled second-moment loss, and physical near-minimizers have a universal strong rescaled profile with angular symmetry. Exactly feasible counterexamples exclude universal higher-moment convergence. At the L1 endpoint the ground-weighted probabilities concentrate at √6 in quadratic Wasserstein distance. At the cap endpoint the preparations approach a filled layer in every finite Lr, while an exact counterexample excludes general strong L∞ convergence. All parameters are fixed in the stated asymptotic limits; classical positivity, convex-moment and coupling methods are explicitly credited.
Read the full paper in the original PDF.
Edition files
- Positive_Heat_Flow_Shells_Scott_Schubr.pdf241.3 KiB · pdf
SHA-256
a79b906042acbf0f33bf661fc8119fe50a71d00af04a29bf927b1efa715dcaec - Positive_Heat_Flow_Shells_Scott_Schubr_Publication_Package.zip278.4 KiB · package
SHA-256
4f5fe611f15c746e8222a4670cc040a27d08e775641a3b678d8f697fc5ee8c15
Available editions
Research details
- Subjects
- Mathematics · Analysis and Spectral Theory
- Keywords
- heat equation, overdetermined boundary flux, spherical shells, positive initial data, convex moment optimization, concentration, Wasserstein stability
- Manuscript date
- 2026-10-22
- Reader reuse
- No additional permission granted
Scope
The results concern tailored nonnegative L2 initial temperatures on a fixed spherical shell with positive inner radius, Dirichlet heat evolution generated by Δ, a normalized ground projection, and one common signed outward derivative across both boundary components. All asymptotic limits fix the shell, dimension, power and positive continuous weight or cap. Quadratic transport and second-moment control do not extend universally to higher transport orders. At the cap endpoint strong convergence holds in every finite Lr, while general strong L∞ convergence fails.
Disclosures
AI-assisted tools were used in research exploration, internal analytic checks, literature comparisons and manuscript preparation.
Reproduction
The complete package contains the 29-page paper PDF, eleven portable LaTeX sources with the complete bibliography, reader build instructions and SHA-256 checksums. The arguments are analytical; no numerical dataset or computational certificate is a premise. The package sources were rebuilt separately with identical rendered pages. Run tectonic main.tex with the compiler resource bundle and fonts available.
Author attribution has been corrected. The mathematical content and edition are unchanged.
Author attribution has been corrected. The mathematical content and edition are unchanged.