发表机构
Institute of Applied Mathematics, Shenzhen Polytechnic University（深圳职业技术学院应用数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文研究单调Sobolev函数的逼近、临界点与水平集，给出平面局部光滑性的肯定答案及高维问题部分肯定与否定答案，提出光滑单调逼近并应用于多个领域。
AI 中文摘要
我们对D. Ntalampekos问题1.7中的平面局部光滑性问题给出了肯定回答，并对其高维问题1.8中的基本逼近和水平集部分给出了肯定和否定回答。在任意维数$n\ge2$中，有界开集上每个$W^{1,p}$中的连续Lebesgue单调函数，对于$1<p<\infty$，都可通过单调$C^{1,\alpha}_{\loc}$函数进行一致且强的$W^{1,p}$逼近，且保持Sobolev边界值不变，$p$-Dirichlet能量不增加。除非原函数是$p$-调和函数，否则能量可以严格减小。在平面情形，我们在每个孤立的$p$-调和临界点处获得光滑局部替换，且$C^1$和Sobolev误差可任意小。梯度指标为$-m$的点可分解为恰好$m$个非退化鞍点。结合Ntalampekos的平面定理，这给出了具有固定边界值的光滑单调密度；在非光滑$p$-调和极小点上，能量增加不可避免但可趋于零。在维数$n\ge3$中，一个显式Lipschitz单调函数在区间内每个水平集上都有非流形点，尽管它允许光滑单调逼近。一个平面齐次$7$-调和函数的乘积扩展也排除了在精确边界和能量约束下的一般离散例外集。因此，问题1.8的答案区分了五个基本逼近性质与更强的拓扑和例外集结论。应用包括约束积分泛函、超水平集的严格$BV$收敛、非线性通量收敛以及在驯顺性假设下持久性图的稳定性。
英文摘要
We give an affirmative answer to the planar local smoothing problem in Question~1.7 of D.~Ntalampekos and positive and negative answers to the basic approximation and level-set parts of his higher-dimensional Question~1.8. In every dimension $n\ge2$, each continuous Lebesgue monotone function in $W^{1,p}$ on a bounded open set admits uniform and strong $W^{1,p}$ approximation by monotone $C^{1,α}_{\loc}$ functions, with unchanged Sobolev boundary values and no increase of the $p$-Dirichlet energy, for $1<p<\infty$. The energy can be made strictly smaller unless the original function is $p$-harmonic. In the plane we obtain smooth local replacement at every isolated $p$-harmonic critical point, with arbitrarily small $C^1$ and Sobolev error. A point of gradient index $-m$ can be resolved into exactly $m$ nondegenerate saddles. Together with Ntalampekos's planar theorem, this gives smooth monotone density with fixed boundary values; at a nonsmooth $p$-harmonic minimizer the energy increase is unavoidable but can tend to zero. In dimensions $n\ge3$, an explicit Lipschitz monotone function has a nonmanifold point on every level in an interval, although it admits smooth monotone approximation. A product extension of a planar homogeneous $7$-harmonic function also rules out a general discrete exceptional set under the exact boundary and energy constraints. The answer to Question~1.8 thus distinguishes the five basic approximation properties from the stronger topological and exceptional-set conclusions. Applications include constrained integral functionals, strict $BV$ convergence of superlevel sets, nonlinear flux convergence, and stability of persistence diagrams under tameness assumptions.
