Thickness functions for elliptic level sets: gradient formulas, normal expansions, and geometric remarks

Abstract

We study the geometry of superlevel sets $\Omega_t = \{u > t\}$ of solutions to $-\Delta u = \mu$ with $\mu \geq 0$ compactly supported in a convex core $C \subset \Omega$. Under the radial monotonicity lemma (due to Shahgholian), each level set is a normal graph over $\partial C$ with thickness function $d_t$. We derive an exact formula for the tangential gradient of $d_t$ and, under a quantitative small-thickness hypothesis $\|d_t\|_{C^1(\partial C)} \ll 1$, an asymptotic expansion of the unit normal to $\Gamma^t = \partial \Omega_t$. We discuss the relation with Shahgholian's theorem and give examples showing that the geometric normal property (GNP) does not imply that $d_t$ is constant, even in the small-thickness regime. This work provides a geometric language for studying elliptic level sets, without claiming a new proof of Shahgholian's theorem.