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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 \(p>1\).

MSC:
49N60 Regularity of solutions in optimal control
35J60 Nonlinear elliptic equations
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References:
[1] CAMPANATO S.: Sistemi ellittici in forma divergenza. Regolarita all’interno. Quaderni, Scuola Norm. Sup., Pisa, 1980. · Zbl 0453.35026
[2] DANECEK J.: Regularity for nonlinear elliptic systems. Comment. Math. Univ. Carolin. 27 (1986), 755-764. · Zbl 0607.35035
[3] GIAQUINTA M.-GIUSTI E.: Differentiability of minima non-differentiate functionals. Invent. Math. 72 (1983), 285-298. · Zbl 0513.49003
[4] GIAQUINTA M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Ann. of Math. Stud. 105, Princenton Univ. Press, Princeton, NJ, 1983. · Zbl 0516.49003
[5] GREVHOLM B.: On the structure of the spaces \({\mathcal L}^{p,\lambda}_k\). Math. Scand. 26 (1970), 189-196. · Zbl 0212.46002
[6] KADLEC J.-NEČAS J.: Sulla regularita delle soluzioni di equazioni ellitiche negli spazi \(H^{k, \lambda}\). Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21 (1967), 527-545. · Zbl 0157.42203
[7] KUFNER A. JOHN O.-FUČÍK S.: Function Spaces. Academia, Prague, 1977.
[8] NEČAS J.: Les méthodes directes en théorie des équations elliptiques. Academia, Prague, 1967. · Zbl 1225.35003
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