Monotone Sobolev Functions in Planar Domains: Level Sets and Smooth Approximation
- PDF / 490,214 Bytes
- 32 Pages / 439.37 x 666.142 pts Page_size
- 94 Downloads / 142 Views
Monotone Sobolev Functions in Planar Domains: Level Sets and Smooth Approximation Dimitrios Ntalampekos Communicated by A. Figalli
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, using the theory of p-harmonic functions, we show that monotone Sobolev functions in planar domains can be approximated uniformly and in the Sobolev norm by smooth monotone functions.
1. Introduction The classical theorem of Sard [38] asserts that for a C 2 -smooth function f in a planar domain almost every value is a regular value. That is, for almost all t ∈ R the set f −1 (t) does not intersect the critical set of f , and hence, f −1 (t) is an embedded 1-dimensional C 2 -smooth submanifold of the plane. This theorem is sharp, in the sense that the C 2 -smoothness cannot be relaxed to C 1 -smoothness, as was shown by Whitney [42]. Here and elsewhere in the paper “1-dimensional” refers to the topological dimension. In fact, Sard’s theorem and some of the other theorems that we quote below have more general statements that hold for maps defined in subsets of Rn , taking values in Rm , and having appropriate regularity. In order to facilitate the comparison to our results, we will only give formulations in the case of real-valued functions defined in planar domains. The author was partially supported by NSF Grant DMS-2000096
D. Ntalampekos
Several generalizations and improvements of Sard’s theorem have been proved since the original theorem was published. In particular, Dubovitskii [9] proved that a C 1 -smooth function f in a planar domain has the property that for almost every value t ∈ R the set f −1 (t) intersects the critical set in a set of Hausdorff 1-measure zero. De Pascale [8] extended the conclusion of Sard’s theorem to Sobolev functions of the class W 2, p , where p > 2. For other versions of Sard’s theorem in the setting of Hölder and Sobolev spaces see [4,5,13,33]. Now, we turn our attention to the structure of the level sets of functions, instead 1, p of discussing the critical set. Theorem 1.6 in [5] states that if f ∈ Wloc (R2 ), then there exists a Borel representative of f such that for almost every t ∈ R the level set f −1 (t) is equal to Z ∪ j∈N K j , where H1 (Z ) = 0, K j ⊂ K j+1 , and K j is contained in an 1-dimensional C 1 -smooth submanifold S j of R2 for j ∈ N. Under increased regularity, Bourgain et al. [6] proved that if f ∈ W 2,1 (R2 ) then for almost every t ∈ R the level set f −1 (t) is an 1-dimensional C 1 -smooth manifold. We
Data Loading...
