Splitting-type variational problems with asymmetrical growth conditions

Abstract

AbstractSplitting-type variational problems $$\begin{aligned} \int _{\Omega }\sum _{i=1}^n f_i(\partial _i w) \, \textrm{d}x\rightarrow \min \end{aligned}$$ ∫ Ω ∑ i = 1 n f i ( ∂ i w ) d x → min with superlinear growth conditions are studied by assuming $$\begin{aligned} h_i(t) \le f''_i(t) \le H_i(t) \qquad (*) \end{aligned}$$ h i ( t ) ≤ f i ′ ′ ( t ) ≤ H i ( t ) ( ∗ ) with suitable functions $$h_i$$ h i , $$H_i$$ H i : $$\mathbb {R}\rightarrow \mathbb {R}^+$$ R → R + , $$i=1$$ i = 1 , ..., n, measuring the growth and ellipticity of the energy density. Here, as the main feature, we do not impose a symmetric behaviour like $$h_i(t)\approx h_i(-t)$$ h i ( t ) ≈ h i ( - t ) and $$H_i(t) \approx H_i(-t)$$ H i ( t ) ≈ H i ( - t ) for large |t|. Assuming quite weak hypotheses on the functions appearing in $$(*)$$ ( ∗ ) , we establish higher integrability of $$|\nabla u|$$ | ∇ u | for local minimizers $$u\in L^\infty (\Omega )$$ u ∈ L ∞ ( Ω ) by using a Caccioppoli-type inequality with some power weights of negative exponent.