Abstract
Let Ω, ⊂R n and n ≥ 4 be even. We show that if a sequence {uj} in W1,n/2(Ω;R n) is almost conformal in the sense that dist (∇uj,R +SO(n)) converges strongly to 0 in Ln/2 and if uj converges weakly to u in W1,n/2, then u is conformal and ∇uj → ∇u strongly in L qloc for all 1 < -q < n/2. It is known that this conclusion fails if n/2 is replaced by any smaller exponent p. We also prove the existence of a quasiconvex function f(A) that satisfies 0 ≤ f(A) ≤ C (1 + ¦A¦n/2) and vanishes exactly onR + SO(n). The proof of these results involves the Iwaniec-Martin characterization of conformal maps, the weak continuity and biting convergence of Jacobians, and the weak-L1 estimates for Hodge decompositions.
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Müller, S., Šverák, V. & Yan, B. Sharp stability results for almost conformal maps in even dimensions. J Geom Anal 9, 671–681 (1999). https://doi.org/10.1007/BF02921978
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DOI: https://doi.org/10.1007/BF02921978