Limits of sequences of volume preserving homeomorphisms in W 1,p, for 0 < p < 1

The paper proves that if H: Ω → SO(d) is Riemann integrable (d ≥ 2, 0 < p < 1), then (Id, H) ∈ M1,p_λ(Ω) by (i) handling piecewise-constant SO(d)-valued H via a smooth cut-off rotation inside disjoint balls (Proposition 4.4), and (ii) approximating general H by dyadic step functions in SO(d) and passing to the limit (Theorem 4.5). The model’s solution uses the same architecture—simple-function (SO(d))-approximation plus a local cut-off-rotation building block and Vitali-style disjoint-ball covering—differing only in the local mechanism (divergence-free flows with skew-symmetric generators rather than the paper’s explicit radial formula). Hence, they are substantially the same proof strategy. Key steps and estimates in the paper that match the model’s plan include the explicit C∞ volume-preserving local gadget (Lemma 3.1 and Proposition 3.2) and the Vitali covering/finite subfamily argument (Lemma 3.4), used to prove Proposition 4.4 and Theorem 4.5 (and the final Lp convergence via (1.3)). See the abstract and Section 4 for the statement and plan, and Lemma 3.1/Proposition 3.2/Lemma 3.4/Proposition 4.4/Theorem 4.5 for the construction and estimates .