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.

MSC:

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

Zbl 0627.49006
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References:

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