Calculus of Variations and Geometric Measure Theory

M. Novaga - E. Paolini

Regularity results for some $1$-homogeneous functionals

created on 26 May 2001
modified on 22 Nov 2002


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

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\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