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Regularity of minima of variational integrals. (English) Zbl 0961.49022
The paper deals with the regularity of derivatives of functions \(u\) minimizing the variational integral \(F(u,\Omega)=\int_\Omega f(x,u,Du) dx\), where \(\Omega \subset \mathbb R^n\), \(n>1\), is an open set, \(u: \Omega \to \mathbb R^N\), \(N>1\), \(Du=\{D_\alpha u^i\}\), \(\alpha =1,\dots ,n\), \(i=1,\dots ,N\) and \(f: \Omega \times \mathbb R^N\times \mathbb R^{nN}\to \mathbb R\) satisfies certain conditions such that either \(Du\in L_{\text{loc}}^{2,n(1-1/p)}(\Omega ,\mathbb R^{nN})\) or \(Du\in \mathcal L^{2,n}_{\text{loc}}(\Omega ,\mathbb R^{nN})\) for some \(p>1\).

49N60 Regularity of solutions in optimal control
35J60 Nonlinear elliptic equations
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