Elliptic Superlevel Sets in the Class $\OO_C$: Boundary-Layer Geometry and Stability of the Return Map

Abstract

We study elliptic superlevel geometry and return dynamics for the class ${\mathcal{O}}_C$. Let $C\subset {\mathbb{R}}^N$ be a compact strictly convex core, and $\Omega \in {\mathcal{O}}_C$ a bounded domain whose boundary is a global normal graph over $\partial C$ with inward normals returning to $\mathrm{int}C$. For $-\Delta u=f$ in $\Omega$ ($u=0$ on $\partial \Omega$), with $f\ge 0$, $f\not\equiv 0$, $\mathrm{supp}f⋐\mathrm{int}C$, we analyze the superlevel sets ${\Omega }_t=\{u>t\}$ for $t\downarrow 0$. We prove that for small $t>0$, $\partial {\Omega }_t$ is a global normal graph given by $\partial {\Omega }_t=\{c+d_t(c)\nu (c):c\in \partial C\}$, satisfying $\Vert d_t-d_0{\Vert }_{C^1}=O(t)$ and $d_t=d_0-t{b}_0+o_{C^1}(t)$, where $b_0$ depends on the Hopf derivative and transversality. Using normal-graph formulas, we show ${\Omega }_t\in {\mathcal{O}}_C$. The return maps $F_t:\partial C\to \partial C$ are well-defined, satisfying $\Vert F_t-F_0{\Vert }_{L^{\infty }}=O(t)$ and $F_t=F_0+t{\mathcal{A}}_0+o_{L^{\infty }}(t)$. For any sequence $t_n\downarrow 0$, the non-autonomous system $x_{n+1}=F_{t_n}(x_n)$ is asymptotically autonomous with background map $F_0$. These results bridge elliptic boundary-layer deformation and return-map dynamics relative to a convex core.