Calculus of Variations and Geometric Measure Theory
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G. Mingione

The Singular Set of Solutions to Non-Differentiable Elliptic Systems

created on 26 Jun 2002
modified on 06 Mar 2003


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


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

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 $n-2$: $$ \mbox{dim}{\mathcal{H}}(\Omega - \Omega0) < n-2\;.$$ We prove that in the general case $0<\alpha <1$, the Hausdorff dimension of the singular set is strictly less than: $n-2\alpha$: $$ \mbox{dim}{\mathcal{H}}(\Omega - \Omega0) < n-2\alpha\;.$$ 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: $$ - \mbox { div } a(x,Du) = b(x,Du)\;.$$

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