Bi-Sobolev mappings with differential matrices in Orlicz Zygmund classes
A homeomorphism \(f:\Omega\to\Omega'\) between open sets in \(\mathbb{R}^n\) is said to be a \(W^{1,p}\)-bi-Sobolev mapping if \(f\in W^{1,p}(\Omega,\Omega')\) and \(f^{-1}\in W^{1,p}(\Omega',\Omega)\). Let \(K_f(x)=|Df(x)|^n/J_f(x)\) if \(J_f(x)\neq 0\), and let \(K_f(x)=1\) otherwise. Here \(J_f\) is the Jacobian of \(f\). The authors investigate the connection between the properties of \(Df^{-1}\) and \(K_f\) for such mappings in \(\mathbb{R}^n\). We cite two of the main theorems. Theorem~1.1. Let \(f:\Omega\to\Omega'=f(\Omega)\) be a \(W^{1,2}\)-bi-Sobolev mapping such that \(|Df^{-1}|\in L^2\log^\alpha L(\Omega')\) for some \(\alpha\geq0\). Then \(K_f\in L\log^\alpha L(\Omega)\), and there exists a nonnegative constant \(c\) depending only on \(\alpha\) such that \[ \int_\Omega K_f(x)\log^\alpha\big(e+K_f(x)\big)dx\leq c\int_{\Omega'}|Df^{-1}(y)|^2\log^\alpha\big(1+|Df^{-1}(y)|\big)dy+\int_\Omega |Df(x)|^2dx. \] Theorem 1.2. Let \(f:\Omega\to\Omega'=f(\Omega)\) be a homeomorphism in \(W^{1,1}\) with finite distortion such that \(K_f\in L\log^\alpha L(\Omega)\) for some \(\alpha\geq0\). Then \(|Df^{-1}|\in L^2\log^2\log_{\text{loc}}L(\Omega')\).
scientific article; zbMATH DE number 5731691