Partial regularity for variational integrals with $(s,\mu ,q)$–Growth

Abstract.

We introduce integrands $f: {\mathbb R}^{nN}\to{\mathbb R}$ of $(s,\mu,q)$–type, which are, roughly speaking, of lower (upper) growth rate $s\geq 1 (q > 1$) satisfying in addition $D^{2}f(Z)\geq \lambda \left(1+|Z|^2\right)^{-\mu/2}$ for some $\mu \in {\mathbb R}$. Then, if $q < 2-\mu +s\frac{2}{n}$, we prove partial $C^{1}$–regularity of local minimizers $u\in W^{1}_{1,loc}(\Omega ,{\mathbb R}^{N})$ by the way including integrands f being controlled by some N–function and also integrands of anisotropic power growth. Moreover, we extend the known results up to a certain limit and present examples which are not covered by the standard theory.