Monotone Sobolev functions in planar domains: level sets and smooth approximation

TL;DR

This paper investigates the structure of level sets of Sobolev functions in planar domains, showing they are mostly simple geometric objects, and proves they can be approximated by smooth monotone functions, extending Sard's theorem.

Contribution

It establishes the topological nature of level sets for Sobolev functions and provides a uniform approximation method by smooth monotone functions in the plane.

Findings

Almost every level set is a point, Jordan curve, or interval homeomorph.

For monotone Sobolev functions, level sets are embedded 1D topological submanifolds.

Monotone Sobolev functions can be uniformly approximated by smooth monotone functions.

Abstract

We prove that almost every level set of a Sobolev function in a planar domain consists of points, Jordan curves, or homeomorphic copies of an interval. For monotone Sobolev functions in the plane we have the stronger conclusion that almost every level set is an embedded $1$-dimensional topological submanifold of the plane. Here monotonicity is in the sense of Lebesgue: the maximum and minimum of the function in an open set are attained at the boundary. Our result is an analog of Sard's theorem, which asserts that for a $C^2$-smooth function in a planar domain almost every value is a regular value. As an application we show that monotone Sobolev functions in planar domains can be approximated uniformly and in the Sobolev norm by smooth monotone functions.