Korn-type inequalities in Orlicz-Sobolev spaces involving the trace-free part of the symmetric gradient and applications to regularity theory
Let \(v: \Omega \to\mathbb R^n\) (\(\Omega \subset\mathbb R^n\)) be a vector field and consider the symmetric gradient \[ \mathcal{E} (v)= \frac{1}{2}( \nabla v+\nabla^T v), \] and the trace-free part \[ \mathcal{E}^D(v)=\mathcal{E} -\frac{1}{n}(\mathrm{Div} v)I. \] \ This article deals with the following variant of Korn's inequality which involves the trace-free part of the gradient of the vector field \(v\): \[ \int_{\Omega} h(|\nabla v|) \,dx \leq \int_{\Omega} h(|\mathcal{E}^D(v)|) \,dx, \] where \(h \in C^2(\mathbb R)\) is an \(N\)-function satisfying some suitable condition and in particular \( h \in \Delta_2 \cap \nabla_2\). An interesting application to the \(C^{1,\alpha}\)-regularity of minimizers of the integral functional \(\int_{\Omega} h(|\mathcal{E}^Dv|) \, dx\) are also given.
scientific article; zbMATH DE number 6099234