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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

MSC:
 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