Partial regularity results for minimizers of quasiconvex functionals of higher order (Q1611400)

The author studies the regularity of minimizers of quasiconvex functionals involving higher order derivatives of the form \(F(u) = \int_\Omega f(D^m u)\). The functional \(F\) is supposed to be uniformly strictly quasiconvex. Using the technique of harmonic approximation the author proves that the minimizers of \(F\) in \(W^{m,p}(\Omega;{\mathbb R}^N)\), with \(p \geq 2\), belong to \(C^{m,\alpha}(\tilde \Omega)\), where \(\tilde \Omega\) is an open subset of \(\Omega\) such that \({\mathcal L}^n(\tilde \Omega \setminus \Omega)=0\). This theorem extends a result by \textit{L. C. Evans} [Arch. Ration. Mech. Anal. 95, 227--252 (1986; Zbl 0627.49006)] obtained in the case \(m=1\) by a blow-up technique.