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.


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