Calculus of Variations and Geometric Measure Theory
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M. Fuchs - G. Mingione

Full $C^{1,\alpha}$-regularity for free and constrained local minimizers of elliptic variational integrals with nearly linear growth

created on 06 Dec 2001
modified on 20 May 2003


Published Paper

Inserted: 6 dec 2001
Last Updated: 20 may 2003

Journal: Manuscripta Math.
Volume: 102
Number: 2
Pages: 227-250
Year: 2000

Preprint No. 617, SFB 256, University of Bonn


We consider local minimizers $u : R^n \supset \Omega \to R^N$ of elliptic variational integrals ${\cal{F}}(u) = \int\limits_{\Omega} f (Du)\, dx$ with integrand f of nearly linear growth. A typical model is: $$ \int
\log (2+\log(2+....+\log(2+
)...))\ dx$$ In the scalar case N = 1 a side condition of the type $u \geq \phi$ may be incorporated (obstacle problem). For $N > 1$ $u$ is an unconstrained minimizer and $f$ is required just to depend on the modulus of $Du$. We show in both cases regularity of minimizers. In particular, in the vectorial case, we show that $u$ has Hölder continuous first derivatives in the interior of the domain $\Omega$; this solves problems raised in papers by Fuchs, Li and Seregin.

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