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

MSC:

 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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References:

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