Monotone Sobolev functions: approximation, critical points, and level sets

Abstract

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.