Abstract
We prove higher integrability and differentiability results for local minimizers u: \({\mathbb {R}^2\supset\Omega\to\mathbb {R}^M}\), M ≥ 1, of the splitting-type energy \({\int_{\Omega}[h_1(|\partial_1 u|)+h_2(|\partial_2 u|)]\,{\rm d}x}\) . Here h 1, h 2 are rather general N-functions and no relation between h 1 and h 2 is required. The methods also apply to local minimizers u: \({\mathbb {R}^2\supset\Omega \to \mathbb {R}^2}\) of the functional \({\int_{\Omega}[h_1(|{\rm div}\,{\rm u}|)+h_2(|\varepsilon^D(u)|)]\,{\rm d}x}\) so that we can include some variants of so-called nonlinear Hencky-materials. Further extensions concern non-autonomous problems.
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Bildhauer, M., Fuchs, M. Differentiability and higher integrability results for local minimizers of splitting-type variational integrals in 2D with applications to nonlinear Hencky-materials. Calc. Var. 37, 167–186 (2010). https://doi.org/10.1007/s00526-009-0257-y
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DOI: https://doi.org/10.1007/s00526-009-0257-y