Sharp Critical Alexandrov Stability and Short-Time Heat Flux
Abstract
Let $\Omega\subset\mathbb R^N$ be a bounded $C^{2,\alpha}$ domain. We first refine the critical quantitative Alexandrov estimate at $p=(N-1)/2$. Writing $H$ for the normalized mean curvature, $H_0=|\partial\Omega|/(N|\Omega|)$, and $\varepsilon_H=\|H-H_0\|_{L^p(\partial\Omega)}$, we prove an annular estimate of order $\varepsilon_H[1+\log^+(C_*/\varepsilon_H)]^{(N-3)/(N-1)}$, uniformly on bounded $C^{2,\alpha}$ geometric classes. This improves the available full-logarithmic critical profile, and logarithmically concentrated perturbations of the sphere show that the exponent \((N-3)/(N-1)\) is optimal. We then apply the geometric estimate to the Dirichlet heat equation with initial temperature 1. A uniform short-time expansion yields $\|H-H_0\|_{L^r}\le C[\delta_r(t)+t^{\alpha/2}]$, where $\delta_r(t)$ is the $L^r$ distance of the boundary heat flux from the constants. Combining this transfer principle with quantitative Alexandrov stability gives a complete sharp heat-flux stability profile: linear in the supercritical regime, fractional-logarithmic at the critical exponent, and power-like in the subcritical regime. In particular, asymptotic $L^r$ constancy of the short-time boundary heat flux forces $\Omega$ to be a ball.
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