AI 中文总结
研究椭圆狄利克雷问题解生成的移动超水平集渐近几何，给出内向单位法向场严格分解，针对薄壳配置形式化离散镜面点反射映射，证明切向位移关系，通过能量表明轨道近似连续梯度流，有限元计算验证收敛速率。
AI 中文摘要
我们研究了由椭圆狄利克雷问题$-\Delta u = f$在区域$\Omega$中的解所生成的移动超水平集$\Omega_t = \{x \in \Omega: u(x) > t\}$的渐近几何。其中非负源$f \not\equiv 0$紧支在严格凸内芯$C \subset \Omega$内。在定量径向单调性条件下，每个边界$\partial\Omega_t$是$\partial C$上的光滑法向图，由厚度函数$d_t \in C^{1,\alpha}(\partial C)$表征，当$t \to 0$时追踪$d_0$。核心贡献是沿水平曲面的内向单位法向场的严格分解：$\mathbf{n}_{\Omega_t} = \nu - \nabla_{\partial C} d_t + \mathcal{G} + \mathcal{P}$，其中$\nu$是静态径向法向，$-\nabla_{\partial C} d_t$是运动驱动向量，$\mathcal{G}$和$\mathcal{P}$是曲率和偏微分方程黑塞余项。对于薄壳配置$(\Vert{}d_0\Vert{}_{C^1} \ll 1)$，我们在$\partial C$上形式化了一个离散镜面点反射映射$F_n$。我们证明切向位移满足$F_n(p) - p = -2d_n(p)\nabla_{\partial C}d_n(p) + R_n(p)$，具有二次控制$\Vert{}R_n\Vert{}_{L^\infty} \le C\Vert{}d_n\Vert{}_{C^1}^2$。使用能量$\mathcal{E}(t) = \int_{\partial C} d_t^2 \, d\mathcal{H}^{N - 1}$，我们表明这些轨道近似由$+\nabla_{\partial C} d_{\tilde{t}}(p)$驱动的连续梯度流到一阶。有限元计算（FEniCS）验证了这些收敛速率。
英文摘要
We investigate the asymptotic geometry of shifting superlevel sets $Ω_t = \{x \in Ω: u(x) > t\}$ generated by solutions to the elliptic Dirichlet problem $-Δu = f$ in $Ω$, where the non-negative source $f \not\equiv 0$ is compactly supported within a strictly convex inner core $C \subset Ω$. Under a quantitative radial monotonicity condition, each boundary $\partialΩ_t$ is a smooth normal graph over $\partial C$ characterized by a thickness function $d_t \in C^{1,α}(\partial C)$ tracking $d_0$ as $t \to 0$.A central contribution is a rigorous decomposition of the inward unit normal field along the level surfaces: $\mathbf{n}_{Ω_t} = ν- \nabla_{\partial C} d_t + \mathcal{G} + \mathcal{P}$, where $ν$ 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.
