## A regularity result for a class of anisotropic systems.(English)Zbl 0909.49026

Consider a minimizer $$u:\Omega\to \mathbb{R}^N$$, $$N\geq 1$$, $$\Omega\subset\mathbb{R}^n$$, $$n\geq 2$$, of the energy $I(u)= \int_\Omega G(\nabla u)dx,$ where $$G$$ is a convex function of class $$C^1$$ satisfying an anisotropic growth condition of the form $\lambda| Q|^q\leq G(Q)\leq \Lambda(1+| Q|^p)$ as well as the ellipticity condition $D^2G(X) (Y,Y)\geq \gamma(1+| X|^2)^{{q-2\over 2}}| Y|^2.$ It is then shown that partial $$C^1$$-regularity holds provided $$p$$ and $$q$$ are related through the inequality $2\leq q\leq p< \min\Biggl\{q+ 1,{q\cdot n\over n-1}\Biggr\}.$

### MSC:

 49N60 Regularity of solutions in optimal control

### Keywords:

regularity; convexity; minimizer