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Massimiliano Morini

Publications and source records attributed to Massimiliano Morini.

At least 37 records · Page 2Linked to original sources
arXiv2017

Existence and uniqueness for anisotropic and crystalline mean curvature flows

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 approximation by suitably regularized problems. The results are valid in any dimension and for arbitrary, possibly unbounded, initial closed sets. The approach accounts for the possible presence of a time-dependent bounded forcing term, with spatial Lipschitz continuity. As a by-product of the analysis, the problem of the convergence of the Almgren-Taylor-Wang minimizing movements scheme to a unique (up to fattening) "flat flow" in the case of general, possibly crystalline, anisotropies is settled.

math.AP↗
arXiv2016

Engineering Curvature Induced Anisotropy in Thin Ferromagnetic Films

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 a significant surface roughness in the film. We show that in general the anisotropy can point in an arbitrary direction depending on the surface curvature. We provide simple examples of these periodic surface structures to demonstrate how to engineer particular anisotropies in the film.

cond-mat.mes-hall↗
arXiv2016

Nonlinear stability results for the modified Mullins-Sekerka and the surface diffusion flow

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-Kawaski energy. In this case, they are exponentially stable for the so-called modified Mullins-Sekerka flow.

math.AP↗
arXiv2016

Existence and uniqueness for a crystalline mean curvature flow

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 global-in-time solution follows via a minimizing movements approach.

math.AP↗
arXiv2014

Nonlocal curvature flows

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, whose first variation is an admissible generalized curvature. Within this class, we implement a minimizing movements scheme and we prove that it approximates the viscosity solution of the corresponding level set PDE. We also describe several examples and applications. Besides recovering and presenting in a unified way existence, uniqueness, and approximation results for several geometric motions already studied and scattered in the literature, the theory developed in this paper allows us to establish also new results.

math.MG↗
arXiv2014

Motion of three-dimensional elastic films by anisotropic surface diffusion with curvature regularization

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 stability in the case of initial data close to a flat configuration are also addressed.

math.AP↗
arXiv2014

Cascade of minimizers for a nonlocal isoperimetric problem in thin domains

For $Ω_\e=(0,\e)\times (0,1)$ a thin rectangle, we consider minimization of the two-dimensional nonlocal isoperimetric problem given by \[ \inf_u E^γ_{Ω_\e}(u)\] where \[ E^γ_{Ω_\e}(u):= P_{Ω_\e}(\{u(x)=1\})+γ\int_{Ω_\e}\abs{\nabla{v}}^2\,dx \] and the minimization is taken over competitors $u\in BV(Ω_\e;\{\pm 1\})$ satisfying a mass constraint $\fint_{Ω_\e}u=m$ for some $m\in (-1,1)$. Here $P_{Ω_\e}(\{u(x)=1\})$ denotes the perimeter of the set $\{u(x)=1\}$ in $Ω_\e$, $\fint$ denotes the integral average and $v$ denotes the solution to the Poisson problem \[ -Δv=u-m\;\mbox{in}\;Ω_\e,\quad\nabla v\cdot n_{\partialΩ_\e}=0\;\mbox{on}\;\partialΩ_\e,\quad\int_{Ω_\e}v=0.\] We show that a striped pattern is the minimizer for $\e\ll 1$ with the number of stripes growing like $γ^{1/3}$ as $γ\to\infty.$ We then present generalizations of this result to higher dimensions.

math.AP↗
arXiv2013

Geometrically induced phase transitions in two-dimensional dumbbell-shaped domains

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 here the case of general non-symmetric dumbbell-shaped domains with a small constriction and general multi-well potentials. Our main results concern the existence and uniqueness of non-constant local minimizers, their full classification in the case of convex bulks, and the complete description of their asymptotic behavior, as the size of the constriction tends to zero.

math.AP↗
arXiv2012

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 $L^1$-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.

math.AP↗
arXiv2012

A Non-Local Mean Curvature Flow and its semi-implicit time-discrete approximation

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 an abstract existence and uniqueness result for viscosity solutions of non-local degenerate Hamiltonians, satisfying suitable continuity assumption with respect to Kuratowsky convergence of the level sets. This abstract setting applies to an approximated flow. Then, by the method of minimizing movements, we also build an "exact" curvature flow, and we illustrate some examples, comparing the results with the standard mean curvature flow.

math.AP↗
arXiv2007

A higher order model for image restoration: the one dimensional case

The one-dimensional version of the higher order total variation-based model for image restoration proposed by Chan, Marquina, and Mulet in [4] is analyzed. 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.

math.OC↗
arXiv2007

A second order minimality condition for the Mumford-Shah functional

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 of the first eigenvalue of a suitable compact operator, the other involving a sort of nonlocal capacity of $Γ$. A sufficient condition for minimality is also deduced. Finally, an explicit example is discussed, where a complete characterization of the domains where the second variation is nonnegative can be given.

math.AP↗
arXiv2007

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.

math.AP↗
arXiv2006

A vanishing viscosity approach to quasistatic evolution in plasticity with softening

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 that, in this case, the use of global minimizers in the corresponding incremental problems is not justified from the mechanical point of view. Thus, we analize a different selection criterion for the solutions of the quasistatic evolution problem, based on a viscous approximation. This leads to a generalized formulation in terms of Young measures, developed in the first part of the paper. In the second part we apply our approach to some concrete examples.

math.AP↗
arXiv2003

Higher order quasiconvexity reduces to quasiconvexity

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 symmetric matrices of a 1-quasiconvex function with the same growth. As a consequence, lower semicontinuity results for second-order variational problems are deduced as corollaries of well-known first order theorems.

math.AP↗