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

Abstract

We study the geometry of superlevel sets $\hat{I}{\text{c◯}}_t=\{u>t\}$ of solutions to $-\hat{I}”u=\hat{I}¼$ with $\hat{I}¼\ge 0$ compactly supported in a convex core $C\subset \hat{I}\text{c◯}$. 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 $\Vert d_t{\Vert }_{C^1(\partial C)}\ll 1$, an asymptotic expansion of the unit normal to $\hat{I}{“}^t=\partial \hat{I}{\text{c◯}}_t$. 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.