Sharp stability results for almost conformal maps in even dimensions

Stefan Müller, Vladimir Šverák, Baisheng Yan

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

Abstract

Let Ω ⊂ Rn and n ≥ 4 be even. We show that if a sequence {uj} in W1,n/2(Ω Rn) 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 Llocq 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 on R+ 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.

Original languageEnglish (US)
Pages (from-to)671-681
Number of pages11
JournalJournal of Geometric Analysis
Volume9
Issue number4
DOIs
StatePublished - 1999

Keywords

  • Quasi-conformal mappings
  • Quasi-convex functions
  • Stability

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