Point-wise characterizations of limits of planar Sobolev homeomorphisms and their quasi-monotonicity
Year: 2025
Type: article
Abstract: We present three novel classifications of the weak sequential (and strong) limits in W 1 , p {W^{1,p}} of planar diffeomorphisms. We introduce a concept called QM condition which is a kind of separation property for pre-images of closed connected sets and show that u satisfies this property exactly when it is the limit of Sobolev homeomorphisms. Further, we prove that u ∈ W id 1 , p ( ( - 1 , 1 ) 2 , ℝ 2 ) {u\in W^{{1,p}}_{\operatorname{id}}((-1,1)^{2},\mathbb{R}^{2})} is the limit of a sequence of homeomorphisms exactly when there are classically monotone mappings g δ : [ - 1 , 1 ] 2 → ℝ 2 {g_{\delta}:[-1,1]^{2}\to\mathbb{R}^{2}} and very small open sets U δ {U_{\delta}} such that g δ = u {g_{\delta}=u} on [ - 1 , 1 ] 2 ∖ U δ {[-1,1]^{2}\setminus U_{\delta}} . Also, we introduce the so-called the three-curve condition, which is an adaption of the NCL condition of [D. Campbell, A. Pratelli and E. Radici, Comparison between the non-crossing and the non-crossing on lines properties, Journal of Mathematical Analysis and Applications 498 2021, Article ID 124956] for u - 1 {u^{-1}} instead of for u, and prove that a map is the
Author: Daniel Campbell
Institution: Charles University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4408919536