Bi-Sobolev solutions to the prescribed Jacobian inequality in the plane with \(L^{p}\) data and applications to nonlinear elasticity
The authors establish existence results in Sobolev spaces \(W^{1,q}(\Omega;\mathbb{R}^2)\) for the Jacobian inequality \[ \det\,\nabla\,\phi\geq f\;\text{in}\;\Omega;\quad \phi=\text{id}\;\text{on}\,\partial \Omega. \] For \(f\geq 0,\;\int_\Omega f\,dx < |\Omega|\) have the solutions the following properties: 1. For every \(f\in L^p(\Omega),\;p>1\) and every \(1<q<(p+1)/2)\) there exists a homeomorphism \(\phi :\bar\Omega \to \bar\Omega\) solving the Jacobian inequality such that \(\phi,\;\phi^{-1}\,\in W^{1,q}(\Omega;\Omega).\) 2. For \(f\in L^\infty(\Omega)\) there exists a bi-Lipschitz solution \(\phi :\bar\Omega \to \bar\Omega\). The bi-Lipschitz regularity is in general sharp. The results are applied to a model functional in nonlinear elasticity, whose integrand features a fast blow up as the Jacobian determinant of the deformation becomes small.
scientific article; zbMATH DE number 6973894