Convex variational integrals with a wide range of anisotropy. - Part I : Regularity results

Abstract

We consider strictly convex energy densities f:\mathbb{R}^{nN}\rightarrow\mathbb{R},f(Z)=g(\left|Z_{1}\right|,...,\left|Z_{n}\right|) if N>1, under non-standard growth conditions. More precisely we assume that for some constants \lambda, \Lambda and for all Z,Y\in\mathbb{R}^{nN} \lambda(1+\left|Z\right|^{2})^{-\frac{\mu}{2}}\left|Y\right|^{2}\leq D^{2}f(Z)(Y,Y)\leq\Lambda(1+\left|Z\right|^{2})^{\frac{q-2}{2}}\left|Y\right|^{2} holds with exponents \mu\in\mathbb{R} and q>1. If u denotes a local minimizer w.r.t. the energy \int f(\nabla w), then we prove L^{q+\varepsilon}-integrability of \left|\nabla u\right| provided that u is locally bounded and q<4-\mu. In particular this is true in the vectorvalued setting and implies partial C^{1,\alpha}-regularity of u together with the additional assumption q<(2-\mu)n/(n-2). In the scalar case we derive local C^{1,\alpha} -regularity from the condition q<4-\mu , again if u is locally bounded. Both results substantially improve what is known up to now (see, for instance, [ELM], [CH], [BF1], [BF2] and the references quoted therein.)