On the distributional Jacobian of maps from SN into SN in fractional Sobolev and Holder spaces

• 成果类型: Article
• 署名作者: Brezis, Haim; Nguyen, Hoai-Minh
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2011.173.2.15
• 发表日期: 2011
• 页码: 1141-1183
• 关键词: weak continuity; Integrability; approximation; determinants; compactness

摘要

H. Brezis and L. Nirenberg proved that if (g(k)) subset of C-0(S-N, S-N) and g is an element of 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 is an element of C-1(S-N, S-N), then Kronecker's formula asserts that deg = 1/vertical bar S-N vertical bar integral(SN) det(del g)d sigma. Consequently, integral(SN) det(del g(k)) d sigma converges to integral(SN) det(del g)d sigma provided g(k) -> g in BMO(S-N). In the same spirit, we consider the quantity J(g, psi) := integral(SN) psi det(del g) d sigma, for all psi is an element of C-1(S-N, R) and study the convergence of J(g(k), psi). In particular, we prove that J(g(k), psi) converges to J(g, psi) for any psi is an element of C-1(S-N, R) if g(k) converges to g in C-0,C-alpha(S-N) for some alpha > N-1/N. Surprisingly, this result is optimal when N > 1. In the case N - 1 we prove that g(k) -> g almost everywhere and lim sup(k ->infinity) vertical bar gk - g vertical bar(BMO) is sufficiently small, then J(g(k), psi) -> J(g, psi) for any psi is an element of C-1(S-1, R). We also establish bounds for J(g, psi) 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.