On the distributional Jacobian of maps from $\mathbb{S}^N$ into $\mathbb{S}^N$ in fractional Sobolev and Hölder spaces

Abstract

H. Brezis and L. Nirenberg proved that if $(g_k) \subset C^0(\mathbb{S}^N, \mathbb{S}^N)$ and $g \in C^0(\mathbb{S}^N, \mathbb{S}^N)$ ($N \ge 1$) are such that $g_k \rightarrow g$ in ${\rm BMO}(\mathbb{S}^N)$, then $\mathrm{deg} \, g_k \rightarrow \mathrm{deg} \, g$. On the other hand, if $g \in C^1(\mathbb{S}^N, \mathbb{S}^N)$, then Kronecker’s formula asserts that $ \mathrm{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 ${\rm BMO}(\mathbb{S}^N)$. In the same spirit, we consider the quantity $ {\bf 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 ${\bf J}(g_k, \psi)$. In particular, we prove that ${\bf J}(g_k, \psi)$ converges to ${\bf J}(g, \psi)$ 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 $\limsup_{k \rightarrow \infty} |g_k – g|_{\rm BMO}$ is sufficiently small, then ${\bf J}(g_k, \psi) \rightarrow {\bf J}(g, \psi)$ for any $\psi \in C^1(\mathbb{S}^1, \mathbb{R})$. We also establish bounds for ${\bf J}(g, \psi)$ 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 $N=1$.