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Quasiconvexity, growth conditions and partial regularity. (English) Zbl 0658.49006
Partial differential equations and calculus of variations, Lect. Notes Math. 1357, 211-237 (1988).
[For the entire collection see Zbl 0648.00008.]
The author considers nondegenerate variational integrals \[ F(u,\Omega):=\int_{\Omega}f(\cdot,u,Du)dx \] being defined for vector functions u: \({\mathbb{R}}^ n\supset \Omega \to {\mathbb{R}}^ N\). The following structural conditions are imposed on the integrand f(x,y,P): \[ \text{smoothness: f is of class \(C^ 2\) w.r.t. P;}\quad (x,y)\to (1+| P|^ m)^{-1}f(x,y,P) \] is uniformly Hölder continuous

\[ \text{growth: }c_ 0| P|^ m\leq f(x,y,P)\leq c_ 1(1+| p|^ m),\quad | F_{pp}(x,y,P)\leq c_ 2(1+| P|^{m- 2}) \] \[ \text{ellipticity: }F_{p^ i_{\alpha}p^ j_{\beta}}(x,y,P) Q^ i_{\alpha} Q^ j_{\beta}\leq c_ 3(\mu +| P|^{m-2})| Q|^ 2. \] Here \(m\geq 2\), \(c_ 0,c_ 1,c_ 2,c_ 3>0\) and \(\mu\geq 0\) denote suitable constants. Nondegeneracy of the integrand means that \(\mu\) has to be different from zero. (A special class of degenerate integrals is treated in the recent paper of M. Giaquinta and G. Modica [Manuscr. Math. 57, 55-99 (1986; Zbl 0607.49003)].) It is then shown that a local minimizer \(u\in H^{1,m}_{loc}(\Omega,{\mathbb{R}}^ N)\) is of class \(C^{1,\alpha}\) on an open subset \(\Omega_ 0\) of \(\Omega\) with \({\mathcal L}^ n(\Omega - \Omega_ 0)=0\). The method of proof is the so-called direct approach based on Caccioppoli type inequalities which are combined with regularity estimates for minimizers of frozen functionals. Moreover, the author gives a survey of related results and discusses the question to what extend the hypothesis concerning the growth and ellipticity properties of the integrand can be weakened.
Reviewer: M.Fuchs

49J45 Methods involving semicontinuity and convergence; relaxation
35D10 Regularity of generalized solutions of PDE (MSC2000)
26B25 Convexity of real functions of several variables, generalizations
35J20 Variational methods for second-order elliptic equations