Positive heat-flow observations on spherical shells: cutoffs, optimal preparation and concentration

Overview

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.

Keywords heat equation · overdetermined boundary flux · spherical shells · positive initial data · convex moment optimization · concentration · Wasserstein stability