Partial regularity of minimizers of quasiconvex integrals with subquadratic growth: the general case (Q2773302)

The authors consider variational integrals of the form NEWLINE\[NEWLINEI(u)=\int_\Omega f(x,u(x),\nabla u(x))dxNEWLINE\]NEWLINE defined for functions \(u :\Omega\to\mathbb{R}^N\) of Sobolev class \(W^{1,p}\), where \(\Omega\) is some domain in \(\mathbb{R}^n\). The integrand \(f\) is a uniformly strictly quasiconvex function having \(p\)-growth with respect to \(\nabla u\), moreover, the dependence on \(x\) and \(u\) is assumed to be sufficiently smooth. As a main feature the paper concentrates on the subquadratic case \(1 < p < 2\) which has been studied by \textit{E. Acerbi} and \textit{N. Fusco} [J. Math. Anal. Appl. 140, No.~1, 115-135 (1989; Zbl 0686.49004)] for integrands just depending on \(\nabla u\). When \(f\) is supposed to depend additionally on \(x\) and \(u\) the technique of Acerbi and Fusco does not apply. The authors overcome this difficulty by combining techniques used in the superquadratic case with new tools developed in the paper [\textit{M. Carozza, N. Fusco, G. Mingione}, Ann. Mat. Pura Appl. (4) 175, 141-164 (1998; Zbl 0960.49025)]. The main result states that even in the subquadratic case local \(I\)-minimizers are of class \(C^{1,\alpha}\) on an open subset of \(\Omega\) with full measure, and this interesting theorem completes the regularity theory for quasiconvex integrals up to a certain extent.