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Regularity for minimizers of non-quadratic functionals: The case \(1<p<2\). (English) Zbl 0686.49004

Hölder continuity for the gradient of minimizers of the functional \(\int | Du|^ pdx\), with u: \(\Omega\) \(\subset {\mathbb{R}}^ n\to {\mathbb{R}}^ N\), is known in the cases [p\(\geq 2\), \(N\geq 1]\) and \([1<p<2\), \(N=1]\), due respectively to K. Uhlenbeck [Acta Math. 138, 219-240 (1977; Zbl 0372.35030)] and E. Di Benedetto [Nonlinear Anal., Theory Methods Appl. 7, 827-850 (1983; Zbl 0539.35027)].
In this paper the authors prove a result of partial \(C^{1,\alpha}\) regularity for minimizers of the functional \(\int f(x,u(x),| Du(x)|)dx\) in the case \(1<p<2\), \(N\geq 1\), where f(x,s,t) is a convex function of t such that \(c_ 1| t|^ p\leq f(x,s,t)\leq c_ 2| t|^ p\).
Reviewer: E.Acerbi

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
35D10 Regularity of generalized solutions of PDE (MSC2000)
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