Massimiliano Morini

Minimality via Second Variation for a Nonlocal Isoperimetric Problem

We discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts. We show that critical configurations with positive second variation are local minimizers of the nonlocal area functional and, in fact, satisfy a quantitative isoperimetric inequality with respect to sets that are L1-close. The link with local minimizers for the diffuse-interface Ohta-Kawasaki energy is also discussed. As a byproduct of the quantitative estimate, we get new results concerning periodic local minimizers of the area functional and a proof, via second variation, of the sharp quantitative isoperimetric inequality in the standard Euclidean case. As a further application, we address the global and local minimality of certain lamellar configurations.

Globally stable quasistatic evolution in plasticity with softening

We study a relaxed formulation of the quasistatic evolution problem in the context of small strain associative elastoplasticity with softening. The relaxation takes place in spaces of generalized Young measures. The notion of solution is characterized by the following properties: global stability at each time and energy balance on each time interval. An example developed in detail compares the solutions obtained by this method with the ones provided by a vanishing viscosity approximation, and shows that only the latter capture a decreasing branch in the stress-strain response.

A HIGHER ORDER MODEL FOR IMAGE RESTORATION: THE ONE-DIMENSIONAL CASE ∗

The higher order total variation-based model for image restoration proposed by Chan, Marquina, and Mulet in [SIAM J. Sci. Comput., 22 (2000), pp. 503–516] is analyzed in one dimension. A suitable functional framework in which the minimization problem is well posed is being proposed, and it is proved analytically that the higher order regularizing term prevents the occurrence of the staircase effect. The generalized version of the model considered here includes, as particular cases, some curvature dependent functionals.

Local calibrations for minimizers of the mumford-shah functional with a regular discontinuity set

Abstract 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 Sw of w is a regular curve connecting two boundary points, then there exists a uniform neighbourhood U of Sw such that w is a minimizer of the Mumford–Shah functional on U with respect to its own boundary conditions on ∂U. 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 on ∂Ω 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 Ω and Sw under which this kind of minimality holds.

Time-dependent systems of generalized Young measures

In this paper some new tools for the study of evolution problems in the framework of Young measures are introduced. A suitable notion of time-dependent system of generalized Young measures is defined, which allows us to extend the classical notions of total variation and absolute continuity with respect to time, as well as the notion of time derivative. The main results are a Helly type theorem for sequences of systems of generalized Young measures and a theorem about the existence of the time derivative for systems with bounded variation with respect to time.

Cascade of minimizers for a nonlocal isoperimetric problem in thin domains

For

Higher-Order Quasiconvexity Reduces to Quasiconvexity

\Omega_\varepsilon=(0,\varepsilon)\times (0,1)

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

a thin rectangle, we consider minimization of the two-dimensional nonlocal isoperimetric problem given by

A VARIATIONAL MODEL FOR INFINITE PERIMETER SEGMENTATIONS BASED ON LIPSCHITZ LEVEL SET FUNCTIONS: DENOISING WHILE KEEPING FINELY OSCILLATORY BOUNDARIES

\inf_u E^{\gamma}_{\Omega_\varepsilon}(u)

Nonlocal Curvature Flows

, where