## A remark on partial regularity of minimizers of quasiconvex integrals of higher order.(English)Zbl 1006.49028

The author studies higher-order variational integrals $I(u)= \int_\Omega f(D^k u) dx$ defined for functions $$u: \mathbb{R}^n\supset \Omega\to\mathbb{R}^N$$ from the Sobolev class $$W^{k,p}(\Omega,\mathbb{R}^N)$$. Here $$p$$ is some exponent from $$(1,\infty)$$, and the integrand $$f$$ is assumed to be a $$C^2$$-function of growth order $$p$$. It is shown that if $$f$$ satisfies the condition of strict quasiconvexity, then local minimizers are of class $$C^{k,\gamma}$$ on an open subset of $$\Omega$$ with full Lebesgue measure. This partial regularity result is achieved through a decay estimate for a suitable excess function.

### MSC:

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