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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
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