Existence and partial regularity in the calculus of variations (Q1105189)

In the first part of the paper the author considers integrals of the form \[ (1)\quad {\mathcal F}(u(\Omega)=\int_{\Omega}f(\cdot,u,Du)dx \] defined for vector valued functions \(u: {\mathbb{R}}^ n\supset \Omega \to {\mathbb{R}}^ N\) and with the additional property that the integrand \(f(x,y,Q)\) satisfies the growth condition \(f_ 0(Q)-g(x)\leq f(x,y,Q)\leq \lambda | Q|^ m+g(x)\) with \(\lambda,g>0\), \(m\geq 2\) and \(f_ 0\) quasiconvex with growth order m. He then introduces a concept of generalized quasi-minima \(u\in H^{1,m}(\Omega,{\mathbb{R}}^ N\)) of the functional (1) and shows \(Du\in L^{m+\epsilon}_{loc}(\Omega,{\mathbb{R}}^{nN})\) for some small \(\epsilon >0\). The proof uses a version of Caccioppoli's inequality being valid for generalized quasi-minima. A second chapter gives some existence theorems for minimizers of quasiconvex functionals (1), a final section is concerned with the partial regularity properties of these minimizers in a special case. Related results can be found for example in a paper by \textit{M. Giaquinta} and \textit{G. Modica} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3, 185-208 (1986; Zbl 0594.49004)].