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On the distributional Jacobian of maps from \(\mathbb{S}^N\) into \(\mathbb{S}^N\) in fractional Sobolev and Hölder spaces. (English) Zbl 1252.58005

Summary: H. Brézis and L. Nirenberg [Sel. Math., New Ser. 1, No. 2, 197–263 (1995; Zbl 0852.58010)] proved that, if \((g_k) \subset C^0(\mathbb{S}^N, \mathbb{S}^N)\) and \(g \in C^0(\mathbb{S}^N, \mathbb{S}^N)\;(N \geq 1)\) are such that \(g_k \rightarrow g\) in \(\text{BMO}(\mathbb{S}^N)\), then \(\deg g_k \rightarrow \deg g\). On the other hand, if \(g \in C^1(\mathbb{S}^N, \mathbb{S}^N)\), then Kronecker’s formula asserts that \( \deg g = \frac{1}{|\mathbb{S}^N|} \int_{\mathbb{S}^N} \det (\nabla g) \, d \sigma\). Consequently, \(\int_{\mathbb{S}^N} \det (\nabla g_k) \, d \sigma\) converges to \(\int_{\mathbb{S}^N} \det (\nabla g) \, d \sigma\) provided \(g_k \rightarrow g\) in \(\text{BMO}(\mathbb{S}^N)\).
In the same spirit, we consider the quantity \(\mathbf {J}(g, \psi) := \int_{\mathbb{S}^N} \psi \det (\nabla g) \, d \sigma\), for all \(\psi \in C^1(\mathbb{S}^N, \mathbb{R}) \) and study the convergence of \(\mathbf{J}(g_k, \psi)\). In particular, we prove that \(\mathbf{J}(g_k, \psi)\) converges to \(\mathbf{J}(g, \psi)\) for any \(\psi \in C^1(\mathbb{S}^N, \mathbb{R})\) if \(g_k\) converges to \(g\) in \(C^{0, \alpha}(\mathbb{S}^N)\) for some \(\alpha > \frac{N-1}{N}\). Surprisingly, this result is “optimal” when \(N > 1\). In the case \(N=1\), we prove that, if \(g_k \rightarrow g\) almost everywhere and \(\limsup_{k \rightarrow \infty} |g_k - g|_{\text{BMO}}\) is sufficiently small, then \(\mathbf{J}(g_k, \psi) \rightarrow \mathbf{J}(g, \psi)\) for any \(\psi \in C^1(\mathbb{S}^1, \mathbb{R})\). We also establish bounds for \(\mathbf{J}(g, \psi)\) which are motivated by the works of J. Bourgain, H. Brezis and H.-M. Nguyen [C. R., Math., Acad. Sci. Paris 340, No. 11, 787–791 (2005; Zbl 1071.55002)] and H.-M. Nguyen [J. Anal. Math. 101, 367–395 (2007; Zbl 1147.47046)]. We pay special attention to the case \(N=1\).

MSC:

58C35 Integration on manifolds; measures on manifolds
58C25 Differentiable maps on manifolds
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
58D15 Manifolds of mappings
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