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Partial regularity results for minimizers of quasiconvex functionals of higher order. (English) Zbl 1010.49023
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 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.

49N60 Regularity of solutions in optimal control
49J45 Methods involving semicontinuity and convergence; relaxation
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