A regularity theorem for minimizers of quasiconvex integrals. (English) Zbl 0627.49007

Let \(\Omega \subset {\mathbb{R}}^ n\) be a bounded set. Let \(f: {\mathbb{R}}^{nN}\to {\mathbb{R}}\) be a function of class \(C^ 2\) such that \[ | f(\xi)| \leq L(1+| \xi |^ p), \]
\[ \int_{\Omega}f(\xi +D\phi)dx\geq \int_{\Omega}[f(\xi)+\gamma (| D\phi |^ 2+| D\phi |^ p)]dx \] for all \(\xi \in {\mathbb{R}}^{nN}\) and all \(\phi \in C^ 1_ 0(\Omega;{\mathbb{R}}^ N)\) (p\(\geq 2\), L, \(\gamma =const>0)\). Then the first main result of the paper under review is as follows. Let \(u\in W^{1,p}(\Omega;{\mathbb{R}}^ N)\) satisfy \[ \int_{\Omega}f(Du)dx\leq \int_{\Omega}f(D(u+\phi))dx,\quad \phi \in W_ 0^{1,p}(\Omega;{\mathbb{R}}^ N). \] Then there exists an open set \(\Omega_ 0\subset \Omega\) such that \(meas(\Omega \setminus \Omega_ 0)=0\) and \(u\in C^{1,\mu}(\Omega_ 0;{\mathbb{R}}^ N)\) for all \(0<\mu <1\). The proof is based on the blow-up method and an approximation lemma combined with a higher integrability result for minima of certain non-coercive functionals. The above result is extended to the case when f also depends on (x,u).
Reviewer: J.Naumann


49J45 Methods involving semicontinuity and convergence; relaxation
35D10 Regularity of generalized solutions of PDE (MSC2000)
35J60 Nonlinear elliptic equations
26B25 Convexity of real functions of several variables, generalizations
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