Global calibrations for the non-homogeneous Mumford-Shah functional
Abstract
Description
Using a calibration method we prove that, if $Γ\subset Ω$ is a closed regular hypersurface and if the function $g$ is discontinuous along $Γ$ and regular outside, then the function $u_β$ which solves $$ \begin{cases} Δu_β=β(u_β-g)& \text{in $Ω\setminusΓ$} \partial_ν u_β=0 & \text{on $\partialΩ\cupΓ$} \end{cases} $$ is in turn discontinuous along $Γ$ and it is the unique absolute minimizer of the non-homogeneous Mumford-Shah functional $$ \int_{Ω\setminus S_u}|\nabla u|^2 dx +{\cal H}^{n-1}(S_u)+β\int_{Ω\setminus S_u}(u-g)^2 dx, $$ over $SBV(Ω)$, for $β$ large enough. Applications of the result to the study of the gradient flow by the method of minimizing movements are shown.
33 pages
33 pages