AI 中文总结
该研究针对梯度依赖半线性方程混合边界连接处的解正则性问题，建立广义共振定理，证明会出现对数异常形态，通过XFEM空间和有限元实验验证了相关结果，揭示了梯度惩罚的共振阻碍效应。
AI 中文摘要
椭圆型偏微分方程解的正则性在混合Dirichlet-Neumann边界连接处会严重退化，经典上以O(r^{1/2})阶主奇异函数为特征。尽管这种线性行为已被充分记录，梯度依赖半线性扰动的引入会改变局部渐近形态。本文证明，与|∇u|线性缩放的梯度惩罚会诱导一个高度局域化的O(r^{-1/2})阶源项，该源项与主齐次微分算子的半整数谱产生共振。这种非正交共振导致标准可分多项式假设在O(r^{3/2})阶失效。我们建立了一个广义共振定理，该定理强制对数异常的出现，提供了由此产生的r^{3/2}ln(r)形态的精确解析形式，以及余项的严格局部Sobolev正则性界。通过计算ℓ₁、ℓ₂、ℓ_∞和任意ℓ_q范数惩罚的精确对数系数和角偏移，我们证明了这种阻碍的普遍性。最后，我们形式化了相应的富集连续Galerkin空间（XFEM），确立了其拟最优性，并给出有限元实验，这些实验证实了预测的局域污染，在无曲率的平坦连接处恢复了ℓ₁、ℓ₂和ℓ_∞惩罚下的预测对数系数，且对连接处的求解恢复了标准有限元求解器中最优的自由度效率；我们最后概述了完全富集实现所需的目标软件架构。
英文摘要
The regularity of solutions to elliptic partial differential equations degrades severely at mixed Dirichlet-Neumann boundary junctions, characterized classically by an $\mathcal{O}(r^{1/2})$ leading singular function. While this linear behavior is well documented, the introduction of gradient-dependent semilinear perturbations alters the local asymptotic profile. This article proves that gradient penalties scaling linearly with $|\nabla u|$ induce a highly localized $\mathcal{O}(r^{-1/2})$ source term that resonates with the half-integer spectrum of the principal homogeneous differential operator. This non-orthogonal resonance causes standard separable polynomial assumptions to fail at order $\mathcal{O}(r^{3/2})$. We establish a generalized resonance theorem that forces the emergence of a logarithmic anomaly, providing the exact analytical formulation of the resulting $r^{3/2} \ln(r)$ profile alongside rigorous local Sobolev regularity bounds for the remainder. By calculating the exact logarithmic coefficients and angular offsets for $\ell_1$, $\ell_2$, $\ell_\infty$, and arbitrary $\ell_q$-norm penalties, we demonstrate the universality of this obstruction. Finally, we formalize the corresponding enriched continuous Galerkin space (XFEM), establish its quasi-optimality, and present finite element experiments that confirm the predicted localized pollution, recover the predicted logarithmic coefficients across the $\ell_1$, $\ell_2$, and $\ell_\infty$ penalties on a curvature-free flat junction, and show that resolving the junction recovers optimal degree-of-freedom efficiency in standard finite element solvers; we close by outlining the targeted software architectures required for a fully enriched implementation.
CommentsWork in progress
