Geometric Gradient Flows from Elliptic Level Sets: Normal Decomposition and Reflection Dynamics

Abstract

We investigate the asymptotic geometry of shifting superlevel sets $\Omega_t = \{x \in \Omega : u(x) > t\}$ generated by solutions to the elliptic Dirichlet problem $-\Delta u = f$ in $\Omega$, where the non-negative source $f \not\equiv 0$ is compactly supported within a strictly convex inner core $C \subset \Omega$. Under a quantitative radial monotonicity condition, each boundary $\partial\Omega_t$ is a smooth normal graph over $\partial C$ characterized by a thickness function $d_t \in C^{1,\alpha}(\partial C)$ tracking $d_0$ as $t \to 0$.A central contribution is a rigorous decomposition of the inward unit normal field along the level surfaces: $\mathbf{n}_{\Omega_t} = \nu - \nabla_{\partial C} d_t + \mathcal{G} + \mathcal{P}$, where $\nu$ is the static radial normal, $-\nabla_{\partial C} d_t$ is the kinematic driving vector, and $\mathcal{G}, \mathcal{P}$ are curvature and PDE Hessian remainder operators.In a thin-shell configuration $(\Vert{}d_0\Vert{}_{C^1} \ll 1)$, we formalize a discrete specular point-reflection mapping $F_n$ on $\partial C$. We prove the tangential displacement satisfies $F_n(p) - p = -2d_n(p)\nabla_{\partial C}d_n(p) + R_n(p)$, with quadratic control $\Vert{}R_n\Vert{}_{L^\infty} \le C\Vert{}d_n\Vert{}_{C^1}^2$. Using the energy $\mathcal{E}(t) = \int_{\partial C} d_t^2 \, d\mathcal{H}^{N-1}$, we show these orbits approximate a continuous gradient flow driven by $+\nabla_{\partial C} d_{\tilde{t}}(p)$ to first order. Finite element computations (FEniCS) validate these convergence rates.