Local calibrations for minimizers of the Mumford-Shah functional with a triple junction

We prove that, if u is a function satisfying all Euler conditions for the Mumford-Shah functional and the discontinuity set of u is given by three line segments meeting at the origin with equal angles, then there exists a neighbourhood U of the origin such that u is a minimizer of the Mumford-Shah functional on U with respect to its own boundary conditions on the boundary of U. The proof is obtained by using the calibration method.Comment: 28 pages, 4 figure

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.Comment: 31 pages, 2 figure

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.Comment: 20 pages. To appear on Rendiconti del Seminario Matematico dell'Universita' di Padov

Finite Difference Approximation of Free Discontinuity Problems

We approximate functionals depending on the gradient of $u$ and on the behaviour of $u$ near the discontinuity points, by families of non-local functionals where the gradient is replaced by finite differences. We prove pointwise convergence, $\Gamma$-convergence, and a compactness result which implies, in particular, the convergence of minima and minimizers.Comment: 39 pages. to appear on Proc. Royal Soc. Edinb. Ser.

Checkpoint proteins come under scrutiny

Details are emerging of the interactions between the kinetochore and various spindle checkpoint proteins that ensure that sister chromatids are equally divided between daughter cells during cell division

The time-dependent von Kármán plate equation as a limit of 3D nonlinear elasticity

The asymptotic behaviour of the solutions of three-dimensional nonlinear elastodynamics in a thin plate is studied, as the thickness h of the plate tends to zero. Under appropriate scalings of the applied force and of the initial values in terms of h, it is shown that three-dimensional solutions of the nonlinear elastodynamic equation converge to solutions of the time-dependent von Kármán plate equation

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 $\Omega$, and the discontinuity set of $w$ is a segment connecting two boundary points, then for every point $(x_0, y_0)$ of $\Omega$ 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 $\partial U$.Comment: 22 pages, 4 figure