Splitting-type variational problems with asymmetrical growth conditions

Splitting-type variational problems $\int_Ω\sum_{i=1}^n f_i(\partial_i w) dx \to \min$ with superlinear growth conditions are studied by assuming $h_i(t) \leq f''_i(t) \leq H_i(t)$ with suitable functions $h_i$, $H_i$: $\mathbb{R} \to \mathbb{R}^+$, $i=1$, $\dots$ , $n$, measuring the growth and ellipticity of the energy density. Here, as the main feature, a symmetric behaviour like $h_i(t)\approx h_i(-t)$ and $H_i(t) \approx H_i(-t)$ for large $|t|$ is not supposed. Assuming quite weak hypotheses as above, we establish higher integrability of $|\nabla u|$ for local minimizers $u\in L^\infty(Ω)$ by using a Caccioppoli-type inequality with some power weights of negative exponent.

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