## Abstract

Let Ω ⊂ R^{n} and n ≥ 4 be even. We show that if a sequence {u^{j}} in W^{1,n/2}(Ω R^{n}) is almost conformal in the sense that dist (∇u^{j}, R^{+}SO(n)) converges strongly to 0 in L^{n/2} and if u^{j} converges weakly to u in W^{1,n/2}, then u is conformal and ∇u^{j} → ∇u strongly in L_{loc}^{q} 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-L^{1} estimates for Hodge decompositions.

Original language | English (US) |
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Pages (from-to) | 671-681 |

Number of pages | 11 |

Journal | Journal of Geometric Analysis |

Volume | 9 |

Issue number | 4 |

DOIs | |

State | Published - 1999 |

## Keywords

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