Point-wise characterizations of limits of planar Sobolev homeomorphisms and their quasi-monotonicity

Abstract

We present three novel classifications of the weak sequential (and strong) limits in 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 ) is the limit of a sequence of homeomorphisms exactly when there are classically monotone mappings g δ : [ - 1 , 1 ] 2 → ℝ 2 and very small open sets U δ such that g δ = u on [ - 1 , 1 ] 2 ∖ U δ . 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 instead of for u, and prove that a map is the W 1 , p limit of planar Sobolev homeomorphisms exactly when it satisfies this property. This improves on results in [G. De Philippis and A. Pratelli, The closure of planar diffeomorphisms in Sobolev spaces, Ann. Inst. H. Poincare Anal. Non Lineaire 37 2020, 181–224] answering the question from [T. Iwaniec and and J. Onninen, Limits of Sobolev homeomorphisms, J. Eur. Math. Soc. (JEMS) 19 2017, 2, 473–505].