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 \(W^{1,n-1}(\Omega,\mathbb{R}^{n})\) is studied. The differential matrix is \(Df=[\partial f^{i}/\partial x_{j}]\) and the corresponding cofactor matrix is \(D^{\sharp}f\). The determinant of the differential matrix \(Df(x)\) is assumed to be nonnegative and the cofactor matrix is assumed to satisfy \(| D^{\sharp}f| ^{n/(n-1)}\in L^{P}(\Omega)\), where \(L^{P}(\Omega)\) is an Orlicz space. The assumptions on \(P\) are \[ \int_{1}^{\infty}{P(s)ds\over s^{2}}=\infty \] and \[ \left[t^{-1}P(t)\right]'\leq 0\leq \left[t^{-s}P(t)\right]',\quad s={n^{2}-2n+1\over n^{2}-n-1}. \] The main result of this paper is that, under the above assumptions, the Jacobian determinant of \(f\) is locally integrable.
scientific article; zbMATH DE number 1777202