# Massimiliano MoriniUniversità di Parma | UNIPR · Dipartimento di Scienze Matematiche Fisiche e Informatiche

Massimiliano Morini

PhD

## About

57

Publications

3,606

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

1,414

Citations

Citations since 2017

## Publications

Publications (57)

We present a characterization of the domain wall solutions arising as minimizers of an energy functional obtained in a suitable asymptotic regime of the micromagnetics for infinitely long thin film ferromagnetic strips in which the magnetization is forced to lie in the film plane. For the considered energy, we provide the existence, uniqueness, mon...

We provide the first general result for the asymptotics of the area preserving mean curvature flow in two dimensions showing that flat flow solutions, starting from any bounded set of finite perimeter, converge with exponential rate to a finite union of equally sized disjoint disks. A similar result is established also for the periodic two-phase Mu...

We consider the flat flow solutions of the mean curvature equation with a forcing term in the plane. We prove that for every constant forcing term the stationary sets are given by a finite union of disks with equal radii and disjoint closures. On the other hand for every bounded forcing term tangent disks are never stationary. Finally in the case o...

In this paper we analyze the Euler implicit scheme for the volume preserving mean curvature flow. We prove the exponential convergence of the scheme to a finite union of disjoint balls with equal volume for any bounded initial set with finite perimeter.

We provide the first general result for the asymptotics of the area preserving mean curvature flow in two dimensions showing that flat flow solutions, starting from any bounded set of finite perimeter, converge with exponential rate to a finite union of equally sized disjoint disks. A similar result is established also for the periodic two-phase Mu...

We present a characterization of the domain wall solutions arising as minimizers of an energy functional obtained in a suitable asymptotic regime of micromagnetics for infinitely long thin film ferromagnetic strips in which the magnetization is forced to lie in the film plane. For the considered energy, we provide existence, uniqueness, monotonicit...

We establish the short-time existence of a smooth solution to the surface diffusion equation with an elastic term and without an additional curvature regularization in three space dimensions. We also prove the asymptotic stability of strictly stable stationary sets.

We consider the flat flow solutions of the mean curvature equation with a forcing term in the plane. We prove that for every constant forcing term the stationary sets are given by a finite union of disks with equal radii and disjoint closures. On the other hand for every bounded forcing term tangent disks are never stationary. Finally in the case o...

In this paper we analyse the Euler implicit scheme for the volume preserving mean curvature flow. We prove the exponential convergence of the scheme to a finite union of disjoint balls with equal volume for any bounded initial set with finite perimeter.

We establish short-time existence of a smooth solution to the surface diffusion equation with an elastic term and without an additional curvature regularization in three space dimensions. We also prove the asymptotic stability of strictly stable stationary sets.

In this paper we prove short-time existence of a smooth solution in the plane to the surface diffusion equation with an elastic term and without an additional curvature regularization. We also prove the asymptotic stability of strictly stable stationary sets.

We investigate the influence of periodic surface roughness in thin ferromagnetic films on shape anisotropy and magnetization behavior inside the ferromagnet. Starting from the full micromagnetic energy and using methods of homogenization and $\Gamma$-convergence we derive a two dimensional local reduced model. Investigation of this model provides a...

We prove existence and uniqueness of weak solutions to anisotropic and crystalline mean curvature flows, obtained as limit of the viscosity solutions to flows with smooth anisotropies.

The large curvature effects on micromagnetic energy of a thin ferromagnetic film with nonlocal dipolar energy are considered. We predict that the dipolar interaction and surface curvature can produce perpendicular anisotropy which can be controlled by engineering a special type of periodic surface shape structure. Similar effects can be achieved by...

An existence and uniqueness result, up to fattening, for crystalline mean curvature flows with forcing and arbitrary (convex) mobilities, is proven. This is achieved by introducing a new notion of solution to the corresponding level set formulation. Such a solution satisfies the comparison principle and a stability property with respect to the appr...

An overview of recent analytical developments in epitaxial growth is presented. The energy release via the onset of dislocations is addressed. Regularity of quasistatic equilibria is studied both in the absence and in the presence of dislocations. Morphological evolution of anisotropic epitaxially strained films is considered under the assumption t...

It is shown that any three-dimensional periodic configuration that is strictly stable for the area functional is exponentially stable for the surface diffusion flow and for the Mullins-Sekerka or Hele-Shaw flow. The same result holds for three-dimensional periodic configurations that are strictly stable with respect to the sharp-interface Ohta-Kawa...

A variational model for epitaxially strained films accounting for the presence of dislocations is considered. Existence, regularity and some qualitative properties of solutions are addressed.

In this paper we review some recent results concerning the following non-local
isoperimetric problem:
where m ∈ (-1,1) is given and prescribes the volume of the two phases
{u = 1} and {u = -1}. Here PΩ stands for the perimeter relative to
Ω (in the sense of Caccioppoli-De Giorgi) and BV(Ω; {-1, 1}) denotes
the space of functions of bounded variatio...

An existence and uniqueness result, up to fattening, for a class of
crystalline mean curvature flows with natural mobility is proved. The results
are valid in any dimension and for arbitrary, possibly unbounded, initial
closed sets. The comparison principle is obtained by means of a suitable weak
formulation of the flow, while the existence of a gl...

In this paper it is shown that any regular critical point of the Mumford-Shah
functional, with positive definite second variation, is an isolated local
minimizer with respect to competitors which are sufficiently close in the
L^1-topology.

Short time existence for a surface diffusion evolution equation with
curvature regularization is proved in the context of epitaxially strained
three-dimensional films. This is achieved by implementing a minimizing movement
scheme, which is hinged on the $H^{-1}$-gradient flow structure underpinning
the evolution law. Long-time behavior and Liapunov...

This paper aims at building a unified framework to deal with a wide class of
local and nonlocal translation-invariant geometric flows. First, we introduce a
class of generalized curvatures, and prove the existence and uniqueness for the
level set formulation of the corresponding geometric flows.
We then introduce a class of generalized perimeters,...

In this paper, we prove short time existence, uniqueness, and regularity for a surface diffusion evolution equation with curvature regularization in the context of epitaxially strained two-dimensional films. This is achieved by using the H
−1-gradient flow structure of the evolution law, via De Giorgi’s minimizing movements. This seems to be the fi...

We obtain a sharp quantitative isoperimetric inequality for nonlocal
$s$-perimeters, uniform with respect to $s$ bounded away from $0$. This allows
us to address local and global minimality properties of balls with respect to
the volume-constrained minimization of a free energy consisting of a nonlocal
$s$-perimeter plus a non-local repulsive inter...

We continue the analysis, started in Morini&Slastikov (2012), of a
two-dimensional non-convex variational problem, motivated by studies on
magnetic domain walls trapped by thin necks. The main focus is on the impact of
extreme geometry on the structure of local minimizers representing the
transition between two different constant phases. We address...

For $\Omega_\e=(0,\e)\times (0,1)$ a thin rectangle, we consider minimization
of the two-dimensional nonlocal isoperimetric problem given by \[ \inf_u
E^{\gamma}_{\Omega_\e}(u)\] where \[ E^{\gamma}_{\Omega_\e}(u):=
P_{\Omega_\e}(\{u(x)=1\})+\gamma\int_{\Omega_\e}\abs{\nabla{v}}^2\,dx \] and
the minimization is taken over competitors $u\in BV(\Omeg...

This contribution describes recent results on a variational approach for the geometric gradient flow of perimeter-like functionals, which include a class of non-local perimeters. In particular, the consistency of the variational approach with viscosity solutions of an appropriate level set equation is established.

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 res...

We consider a variational model introduced in the physical literature to describe the epitaxial growth of an elastic film
over a thick flat substrate when a lattice mismatch between the two materials is present. We study quantitative and qualitative
properties of equilibrium configurations, that is, of local and global minimizers of the free-energy...

We address in this paper the study of a geometric evolution, corresponding to
a curvature which is non-local and singular at the origin. The curvature
represents the first variation of the energy recently proposed as a variant of
the standard perimeter penalization for the denoising of nonsmooth curves.
To deal with such degeneracies, we first give...

We prove that a suitable rescaling of biased Perona–Malik energies, defined in the discrete setting, Γ-converges to an anisotropic version of the Mumford–Shah functional. Numerical results are discussed.

Recently Frank and Seiringer have shown an isoperimetric inequality for nonlocal perimeter functionals arising from Sobolev seminorms of fractional order. This isoperimetric inequality is improved here in a quantitative form.

We address the effect of extreme geometry on a non-convex variational problem, motivated by studies on magnetic domain walls trapped by thin necks. The recent analytical results of uc(Kohn and Slastikov) (Calc. Var. Partial Differ. Equ. 28:33-57, 2007) revealed a variety of magnetic structures in three-dimensional ferromagnets depending on the size...

In this paper the reconstruction of damaged piecewice constant color images is studied using a RGB total variation based model for colorization/inpainting. In particular, it is shown that when color is known in a uniformly distributed region, then reconstruction is possible with maximal fidelity.

We propose a new model for segmenting piecewise constant images with irregular
object boundaries: a variant of the Chan–Vese model [T. F. Chan and L. A. Vese, IEEE Trans.
Image Process., 10 (2000), pp. 266–277], where the length penalization of the boundaries is replaced
by the area of their neighborhood of thickness ε. Our aim is to keep fine deta...

We consider a variational model introduced in the physical literature to de-scribe the epitaxial growth of an elastic film over a thick flat substrate when a lattice mismatch between the two materials is present. We prove existence of minimizing configurations, study their regularity properties, and establish sev-eral quantitative and qualitative p...

A new necessary minimality condition for the Mumford-Shah functional is derived by means of second order variations. It is
expressed in terms of a sign condition for a nonlocal quadratic form on H
1
0(Γ), Γ being a submanifold of the regular part of the discontinuity set of the critical point. Two equivalent formulations are provided:
one in terms...

Strained epitaxial films grown on a relatively thick substrate are considered in the context of plane linear elasticity. The
total free energy of the system is assumed to be the sum of the energy of the free surface of the film and the strain energy.
Because of the lattice mismatch between film and substrate, flat configurations are in general ener...

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 t...

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 interva...

The role of surfactants in stabilizing the formation of bubbles in foams is studied using a phase-field model. The analysis
is centered on a van der Walls–Cahn– Hilliard-type energy with an added term which accounts for the interplay between the
presence of a surfactant density and the creation of interfaces. In particular, it is concluded that the...

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 to extend the classical notions of total variation and absolute continuity with respect to time, as well as the notion of time derivativ...

We prove necessary and sufficient conditions for the validity of the classical chain rule in the Sobolev space W-loc(1,1)(R-N; R-d) and in the space BVloc(R-N; R-d) of functions of bounded variation.

We deal with quasistatic evolution problems in plasticity with softening, in the framework of small strain associative elastoplasticity. The presence of a nonconvex term due to the softening phenomenon requires a nontrivial extension of the variational framework for rate-independent problems to the case of a nonconvex energy functional. We argue th...

We present a general theory to study optimal regularity for a large class of nonlinear elliptic systems satisfying general boundary conditions and in the presence of a geometric transmission condition on the free boundary. As an application we give a full positive answer to a conjecture of De Giorgi on the analyticity of local minimizers of the Mum...

Following the $\Gamma$-convergence approach introduced by Müller and Ortiz, the convergence of discrete dynamics for Lagrangians with quadratic behavior is established.

We study a one-dimensional variational problem with two or more level set constraints. The existence of global and local
minimizers turns out to be dependent on the regularity of the energy density. A complete characterization of local minimizers
and the underlying energy landscape is provided. The Γ -limit when the phases exhaust the whole domain...

In this paper it is shown that higher order quasiconvex functions suitable in the variational treatment of problems involving second derivatives may be extended to the space of all matrices as classical quasiconvex functions. Precisely, it is proved that a smooth strictly 2-quasiconvex function with p-growth at infinity, p>1, is the restriction to...

In this paper we give a partial answer to a conjecture of De Giorgi, namely we prove that in dimension two the regular part of the discontinuity set of a local minimizer of the homogeneous Mumford–Shah functional is analytic with the exception of at most a countable number of isolated points.

We prove that a wide class of singularly perturbed functionals generates as Γ-limit a functional related to a free-discontinuity problem. Several applications of the result are shown.

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 $$ \begin{cases} \Delta u_{\beta}=\beta(u_{\beta}-g)& \text{in $\Omega\setminus\Gamma$} \partial_{\nu} u_{\beta}=0 & \text{on $\p...

Using a calibration method, we prove that, if omega 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 omega 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 tha...

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.

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...

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....