Calculus of Variations and Geometric Measure Theory

S. Conti - C. De Lellis

Sharp upper bounds for a variational problem with singular perturbation

created by delellis on 22 Jun 2005
modified on 03 May 2011


Published Paper

Inserted: 22 jun 2005
Last Updated: 3 may 2011

Journal: Math. Ann.
Volume: 338
Number: 1
Pages: 119-146
Year: 2007


Let $\Omega$ be a $C^2$ bounded open set of $*R*^2$ and consider the functionals $$ F\varepsilon (u)\;:=\; \int\Omega \left\{\frac{(1-
\nabla u (x)
2)2}{\varepsilon} + \varepsilon
D2 u (x)
2\right\}\, dx\, . $$ We prove that if $u\in W^{1, \infty} (\Omega)$, $
\nabla u
=1$ a.e., and $\nabla u\in BV$, then $$ \Gamma-\lim{\varpesilon\downarrow0} F\varepsilon (u)=\frac13 \int{J{\nabla u}}
\nabla u
3 d\mathcal{H}1 \, . $$ The new result is the $\Gamma-\limsup$ inequality.

For the most updated version and eventual errata see the page