arXiv subjects

Massimiliano Morini

Publications and source records attributed to Massimiliano Morini.

40 records · Page 3Linked to original sources

arXiv2001

Global calibrations for the non-homogeneous Mumford-Shah functional

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.

math.FA↗

arXiv2001

Local calibrations for minimizers of the Mumford-Shah functional with a regular discontinuity set

Using a calibration method, we prove that, if w is a function which satisfies all Euler conditions for the Mumford-Shah functional on a two-dimensional domain, and the discontinuity set S of w is a regular curve connecting two boundary points, then there exists a uniform neighbourhood U of S such that w is a minimizer of the Mumford-Shah functional on U with respect to its own boundary conditions. We show that Euler conditions do not guarantee in general the minimality of w in the class of functions with the same boundary value of w and whose extended graph is contained in a neighbourhood of the extended graph of w, and we give a sufficient condition in terms of the geometrical properties of the domain and the discontinuity set under which this kind of minimality holds.

math.FA↗

arXiv2000

Local calibrations for minimizers of the Mumford-Shah functional with rectilinear discontinuity sets

Using a calibration method, we prove that, if $w$ is a function which satisfies all Euler conditions for the Mumford-Shah functional on a two-dimensional open set $Ω$, and the discontinuity set of $w$ is a segment connecting two boundary points, then for every point $(x_0, y_0)$ of $Ω$ there exists a neighbourhood $U$ of $(x_0, y_0)$ such that $w$ is a minimizer of the Mumford-Shah functional on $U$ with respect to its own boundary values on $\partial U$.

math.FA↗

arXiv2000

Functionals depending on curvatures with constraints

We deal with a family of functionals depending on curvatures and we prove for them compactness and semicontinuity properties in the class of closed and bounded sets which satisfy a uniform exterior and interior sphere condition. We apply the results to state an existence theorem for the Nitzberg and Mumford problem under this additional constraint.

math.FA↗