Abstract:We consider local minimizers $u : \Bbb R^2\supset \Omega \to \Bbb R^N$ of variational integrals like $\int _\Omega [(1+|\partial _1 u|^{2})^{p/2}+(1+|\partial _2 u|^{2})^{q/2}] dx$ or its degenerate variant $\int _\Omega [|\partial _1 u|^p+|\partial _2 u|^q] dx$ with exponents $2\leq p < q < \infty $ which do not fall completely in the category studied in Bildhauer M., Fuchs M., Calc. Var. {16} (2003), 177--186. We prove interior $C^{1,\alpha }$- respectively $C^{1}$-regularity of $u$ under the condition that $q < 2p$. For decomposable variational integrals of arbitrary order a similar result is established by the way extending the work Bildhauer M., Fuchs M., Ann. Acad. Sci. Fenn. Math. {31} (2006), 349--362.
Keywords: non-standard growth, vector case, local minimizers, interior regularity, problems of higher order
AMS Subject Classification: 49N60, 35J50, 35J35