H. Brezis and L. Nirenberg proved that if (gk) ⊂ C0 (SN,SN) and g ∈ C0 (SN,SN) (N ≥ 1) are such that gk → g in BMO(SN), then deg gk →deg g. On the other hand, if g ∈ C1 (SN,SN), then Kronecker's formula asserts that deg. Consequently, converges to provided gk → g in BMO(SN). In the same spirit, we consider the quantity, for all ψ ∈ C1 (SN,R) and study the convergence of J(gk,ψ). In particular, we prove that J(gk,ψ) converges to J(g,ψ) for any ψ ∈ C1 (SN,R) if gk converges to g in C0,α(SN) for some α >. Surprisingly, this result is "optimal" when N > 1. In the case N = 1 we prove that if gk → g almost everywhere and lim supk→∞|gk - g|BMO is sufficiently small, then J(gk,ψ) → J(g,ψ) for any ψ ∈ C1 (S1, 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.
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Statistics, Probability and Uncertainty