Published Paper
Inserted: 11 may 2001
Last Updated: 9 jun 2004
Journal: Annali dela Scuola Normale Superiore
Pages: 34
Year: 2002
Abstract:
Using a calibration method we prove that, if $\Gamma\subset \Omega$ is a
closed regular hypersurface and if the function $g$ is
discontinuous along $\Gamma$ and regular outside, then the
function $u_{\beta}$ which solves
$$\Delta u{\beta}=\beta(u{\beta}-g) $$
in $\Omega\setminus\Gamma$ with homogeneous Neumann boundary conditions,
is in turn discontinuous along $\Gamma$ and it is the unique
absolute minimizer of the non-homogeneous Mumford-Shah functional
$$\int{\Omega\setminus Su}
\nabla u
2\, dx +{\cal H}{n-1}(Su)+\beta\int{\Omega\setminus Su}(u-g)2\, dx,$$
over $SBV(\Omega)$, for $\beta$ large enough. Applications of the
result to the study of the gradient flow by the method of minimizing movements
are shown.
Keywords: minimizing movements, calibration method
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