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On the distributional Jacobian of maps from $\mS^N$ into $\mS^N$ in fractional Sobolev and Hölder spaces

Haïm Brezis, Hoai-Minh Nguyen
2011 Annals of Mathematics  
H. Brezis and L. Nirenberg proved that if (g k ) ⊂ C 0 (S N , S N ) and g ∈ C 0 (S N , S N ) (N ≥ 1) are such that g k → g in BMO(S N ), then deg g k → deg g. On the other hand, if g ∈ C 1 (S N , S N ), then Kronecker's formula asserts that deg g = 1 In the same spirit, we consider the quantity J(g, ψ) := S N ψ det(∇g) dσ, for all ψ ∈ C 1 (S N , R) and study the convergence of J(g k , ψ). In particular, we prove that J(g k , ψ) converges to J(g, ψ) for any ψ ∈ C 1 (S N , R) if g k converges to
more » ... f g k converges to g in C 0,α (S N ) for some α > N −1 N . Surprisingly, this result is "optimal" when N > 1. In the case N = 1 we prove that if g k → g almost everywhere and lim sup k→∞ |g k − g|BMO is sufficiently small, then J(g k , ψ) → J(g, ψ) for any ψ ∈ C 1 (S 1 , R). We also establish bounds for J(g, ψ) which are motivated by the works of J. Bourgain, H. Brezis, and H.-M. Nguyen and H.-M. Nguyen. We pay special attention to the case N = 1.
doi:10.4007/annals.2011.173.2.15 fatcat:gdpfhmgop5gtffuqtmbs6olhyq