\({\mathcal H}^1\)-estimates of Jacobians by subdeterminants
Let \(\Omega\) be a nonempty open subset in \(\mathbb R^n\), \(n\geq 2\). A mapping \(f=(f^{1},\dots,f^{2}):\Omega\to\mathbb R^n\) in the Sobolev space \(W^{1,n-1}_{\text{loc}}(\Omega,\mathbb R^n)\) is studied. The differential matrix is \(Df=[\partial f^{i}/\partial x_{j}]\) and its cofactor matrix is \(D^{\sharp}f\). The cofactor matrix is assumed to satisfy that \(| D^{\sharp}f| \in L^{n/(n-1)}(\Omega)\). The main result of this paper is that the Jacobian determinant \(\det Df(x)\) belongs to the Hardy space \({\mathcal H}^{1}(\Omega)\) and that the following uniform estimate holds \[ \| \det Df\| _{{\mathcal H}^{1}(\Omega)}\leq C(n) \int_{\Omega}| D^{\sharp}f(x)| ^{n/(n-1)}\,dx. \] This theorem answers a question raised by \textit{S. Müller}, \textit{T. Qi} and \textit{B. S. Yan} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 11, No. 2, 217--243 (1994; Zbl 0863.49002)].
scientific article; zbMATH DE number 1837401