A modification of the blowup technique for variational integrals with subquadratic growth (Q1363586)

In the paper variational integrals in the form \(J(u)=\int_\Omega f(Du) dx\) are considered, where \(\Omega\subset{\mathbb R}^n\) is a bounded domain, \(u\) is a vector valued function in the Sobolev class \(H^{1,p}(\Omega,{\mathbb R}^N)\) and \(f:{\mathbb R}^{nN}\rightarrow {\mathbb R}\) is a strictly convex function of class \(C^2\) which satisfies a \(p\)-growth condition for some \(1<p<2\). The authors prove some partial regularity results for local minimizers of \(J\). Such results are also contained in a paper by \textit{G. Anzellotti} and \textit{M. Giaquinta} [Arch. Ration. Mech. Anal. 102, No. 3, 243-272 (1988; Zbl 0658.49005)], but they use a different technique which is based on blowup arguments. When \(n=2\) also an everywhere \(C^{1,\alpha}\)-regularity for local minimizers is proved under assumptions which are more general with respect to known results.