*Published Paper*

**Inserted:** 26 jun 2002

**Last Updated:** 6 mar 2003

**Journal:** Arch. Ration. Mech. Anal.

**Volume:** 166

**Number:** 4

**Pages:** 287-301

**Year:** 2003

**Abstract:**

For a nonlinear elliptic system of the type:
$$ - \mbox { div } a(x,Du) = 0\;,$$
it is know that if the vector field $a$ is $\alpha$-Holder continuous with respect to the variable $x$ then any weak solution is $C^{1,\alpha}$ partially regular: $u \in C^{1,\alpha}(\Omega_0)$ where $\Omega_0$ is an open subset with full measure: $

\Omega - \Omega_0

=0$.

No estimate for the singular set $\Omega - \Omega_0$ is known, but when $\alpha=1$ (differentiability of the system). In this case the Hausdorff Dimension of $\Omega - \Omega_0$ is strictly less than $n-2$:
$$ \mbox{dim}_{{\mathcal{H}}}(\Omega - \Omega_{0)} < n-2\;.$$
We prove that in the general case $0<\alpha <1$, the Hausdorff dimension of the singular set is strictly less than: $n-2\alpha$:
$$ \mbox{dim}_{{\mathcal{H}}}(\Omega - \Omega_{0)} < n-2\alpha\;.$$
This is the first result valid under the natural assumptions of Holder continuity, to which the classical partial regularity theory applies.

The result extends to the case of systems of the type: $$ - \mbox { div } a(x,Du) = b(x,Du)\;.$$