Recent zbMATH articles in MSC 26B35https://www.zbmath.org/atom/cc/26B352021-04-16T16:22:00+00:00WerkzeugMonotone Sobolev functions in planar domains: level sets and smooth approximation.https://www.zbmath.org/1456.260152021-04-16T16:22:00+00:00"Ntalampekos, Dimitrios"https://www.zbmath.org/authors/?q=ai:ntalampekos.dimitriosThe aim of the present work is to describe the structure of the level sets of Sobolev functions defined in a planar domain. The author establishes that almost every level set of a Sobolev function in a planar domain consists of points, Jordan curves, or homeomorphic copies of an interval. Furthermore, a stronger result is proved in the case that the Sobolev function in the plane is monotone in the Lebesgue sense (i.e., the maximum and minimum of the function in an open set are attained at the boundary), namely almost every level set is an embedded 1-dimensional (in the topological sense) submanifold of the plane. The obtained result is an analog of Sard's theorem for \(C^2\)-smooth functions in a planar domain, which asserts that almost every value is a regular value. Using the theory of \(p\)-harmonic functions, as an application of the obtained results, he proves that monotone Sobolev functions in planar domains can be approximated uniformly and in the Sobolev norm by smooth monotone functions.
Reviewer: Andrey Zahariev (Plovdiv)