Differentiability and higher integrability results for local minimizers of splitting-type variational integrals in 2D with applications to nonlinear Hencky-materials

Abstract

We prove higher integrability and differentiability results for local minimizers u:\mathbb{R}^{2}\supset\Omega\rightarrow\mathbb{R}^{M}, M\geq, of the splitting-type energy \int_{\Omega}\left[h_{1}(\left|\partial_{1}u\right|)+h_{2}(\left|\partial_{2}u\right|)\right]dx. Here h_{1}, h_{2} are rather general N-functions and no relation between hh_{1} and h_{2} is required. The methods also apply to local minimizers u:\mathbb{R}^{2}\supset\Omega\rightarrow\mathbb{R}^{2} of the functional \int_{\Omega}\left[h_{1}(\left|\textrm{div}u\right|)+h_{2}(\left|\varepsilon^{D}(u)\right|)\right]dx so that we can include some variants of so-called nonlinear Hencky-materials. Further extensions concern non-autonomous problems.