an:04075922
Zbl 0658.49006
Giaquinta, Mariano
Quasiconvexity, growth conditions and partial regularity
EN
Partial differential equations and calculus of variations, Lect. Notes Math. 1357, 211-237 (1988).
1988
a
49J45 35D10 26B25 35J20
variational integrals; smoothness; growth; ellipticity; local minimizer; Caccioppoli type inequalities; regularity estimates
[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 H??lder 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 \textit{M. Giaquinta} and \textit{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.
M.Fuchs
Zbl 0648.00008; Zbl 0607.49003