Prescribing the Jacobian in critical spaces
The topic of this article is the study of manifold-valued Sobolev mappings. This setting differs from that of real-valued Sobolev mappings. There are for example circle-valued mappings in the Sobolev space \(W^{1,1}\) that cannot be approximated in \(W^{1,1}\) with smooth circle-valued mappings. Moreover, singularities occur in the regularity theory of manifold-valued maps. \textit{H. Brézis} et al. [Commun. Math. Phys. 107, 649--705 (1986; Zbl 0608.58016)] discovered the possibility of detecting the singularities of maps via the distributional Jacobian. The authors of the paper under review suggest the following program in the Sobolev space \(W^{s,p}(\Omega;\mathbb{S}^{k-1})\) with \(s>0\), \(1\leq p<\infty\) and \(\Omega\) an open set in \(\mathbb{R}^m\) or an \(m\)-dimensional manifold: Task 1: Prove the density in \(W^{s,p}(\Omega;\mathbb{S}^{k-1})\) of a suitable class \(\mathcal{R}\) of \(\mathbb{S}^{k-1}\)-valued maps \(u\) smooth except on a small set \(A(u)\). Task 2: Define the distributional Jacobian \(Ju\) of a map \(u\in W^{s,p}(\Omega;\mathbb{S}^{k-1})\). Task 3: Determine the range of the map \(W^{s,p}(\Omega;\mathbb{S}^{k-1})\ni u\mapsto Ju\). The paper under review contributes to the Tasks~\(2\) and \(3\) for certain exponents. The authors give a definition for a distributional Jacobian \(J\) and describe the images of Sobolev functions under \(J\) in terms of (boundaries of) currents. The authors start with three chapters presenting the main topics and technical tools needed (Sobolev spaces of currents and forms, rectifiability, area, coarea, and interpolation inequalities). The next section contains the verification (up to the proof of a trace-lemma that is deferred to the end of the paper) that the proposed definition of the Jacobian is sound and that its range lies in the family of boundaries of currents. The following section then treats the surjectivity of \(J\).
scientific article; zbMATH DE number 6345762
scientific article