*Published Paper*

**Inserted:** 26 may 2001

**Last Updated:** 22 nov 2002

**Journal:** Nonlinear Analysis: Real World Applications

**Volume:** 3

**Number:** 4

**Pages:** 555-566

**Year:** 2002

**Notes:**

available on http:/www.sciencedirect.com

**Abstract:**

\def{\loc}{\mathrm{loc}}\def{\R}{\mathbf R} We consider local minimizers for a class of $1$-homogeneous integral functionals defined on $BV_\loc(\Omega)$, with $\Omega\subset \R^2$. Under general assumptions on the functional, we prove that the boundary of the subgraph of such minimizers is (locally) a lipschitz graph in a suitable direction. The proof of this statement relies on a regularity result holding for boundaries in $\R^2$ which minimize an anisotropic perimeter. This result is applied to the boundary of sublevel sets of a minimizer $u\in BV_\loc(\Omega)$.

We also provide an example which shows that such regularity result is optimal.

**Keywords:**
regularity, crystals, anisotropy

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