*Published Paper*

**Inserted:** 6 dec 2001

**Last Updated:** 20 may 2003

**Journal:** Manuscripta Math.

**Volume:** 102

**Number:** 2

**Pages:** 227-250

**Year:** 2000

**Notes:**

Preprint No. 617, SFB 256, University of Bonn

**Abstract:**

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

Du

\log (2+\log(2+....+\log(2+

Du

)...))\ 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 HÃ¶lder continuous first derivatives in the interior of the domain $\Omega$;
this solves problems raised in papers by Fuchs, Li and Seregin.