arXiv subjects
Publications and source records attributed to Massimiliano Morini.
In \cite{CMP17} a novel distributional approach has been introduced to provide a well-posed formulation of a class of crystalline mean curvature flows. In this paper, such an approach is extended to the nonlocal setting. Applications include the fractional mean curvature flow and the Minkowski flow; i.e., the geometric flow generated by the $(N-1)$-dimensional Minkowski pre-content.
We consider a fully discrete and explicit scheme for the mean curvature flow of boundaries, based on an elementary diffusion step and a precise redistancing operation. We give an elementary convergence proof for the scheme under the standard CFL condition $h\sim\e^2$, where $h$ is the time discretization step and $\e$ the space step. We discuss extensions to more general convolution/redistancing schemes.
We study the isoperimetric problem for capillary surfaces with a general contact angle $θ\in (0, π)$, outside convex infinite cylinders with arbitrary two-dimensional convex section. We prove that the capillary energy of any surface supported on any such convex cylinder is strictly larger than that of a spherical cap with the same volume and the same contact angle on a flat support, unless the surface is itself a spherical cap resting on a facet of the cylinder. In this class of convex sets, our result extends for the first time the well-known Choe-Ghomi-Ritoré relative isoperimetric inequality, corresponding to the case $θ= π/2$, to general angles.
We consider here a fully discrete variant of the implicit variational scheme for mean curvature flow [AlmTayWan,LucStu], in a setting where the flow is governed by a crystalline surface tension defined by the limit of pairwise interactions energy on the discrete grid. The algorithm is based on a new discrete distance from the evolving sets, which prevents the occurrence of the spatial drift and pinning phenomena identified in [MisiatsYip16,BraGelNov] in a similar discrete framework. We provide the first rigorous convergence result holding in any dimension, for any initial set and for a large class of purely crystalline anisotropies, in which the spatial discretization mesh can be of the same order or coarser than the time step.
We study the asymptotic behavior of the volume preserving mean curvature and the Mullins-Sekerka flat flow in three dimensional space. Motivated by this we establish a 3D sharp quantitative version of the Alexandrov inequality for $C^2$-regular sets with a perimeter bound.
A comparison theorem by Choe, Ghomi and Ritoré states that the exterior isoperimetric profile $I_\mathcal{C}$ of any convex body $\mathcal{C}$ in $\mathbb{R}^N$ lies above that of any half-space $H$. We characterize convex bodies such that $I_\mathcal{C}\equiv I_H$ in terms of a notion of "maximal affine dimension at infinity'', briefly called the asymptotic dimension $d^*(\mathcal{C})$ of $\mathcal{C}$. More precisely, we show that $I_\mathcal{C}\equiv I_H$ if and only if $d^*(\mathcal{C})\ge N-1$. We also show that if $d^*(\mathcal{C})\le N-2$, then, for large volumes, $I_\mathcal{C}$ is asymptotic to the isoperimetric profile of $\mathbb{R}^N$. We then estimate, in terms of $d^*(\mathcal{C})$-dependent power laws, the order as $v\to\infty$ of the difference between $I_\mathcal{C}$ and the isoperimetric profile of $\mathbb{R}^N$.
In this paper we address anisotropic and inhomogeneous mean curvature flows with forcing and mobility, and show that the minimizing movements scheme converges to level set/viscosity solutions and to distributional solutions \textit{à la} Luckhaus-Sturzenhecker to such flows, the latter holding in low dimension and conditionally to a convergence of the energies. By doing so we generalize recent works concerning the evolution by mean curvature by removing the hypothesis of translation invariance, which in the classical theory allows to simplify many arguments.
In this paper we introduce the notion of parabolic $α$-Riesz flow, for $α\in(0,d)$, extending the notion of $s$-fractional heat flows to negative values of the parameter $s=-\fracα{2}$. Then, we determine the limit behaviour of these gradient flows as $α\to 0^+$ and $α\to d^-$. To this end we provide a preliminary $Γ$-convergence expansion for the Riesz interaction energy functionals. Then we apply abstract stability results for uniformly $λ$-convex functionals which guarantee that $Γ$-convergence commutes with the gradient flow structure.
In this paper we study the regularity properties of $Λ$-minimizers of the capillarity energy in a half space with the wet part constrained to be confined inside a given planar region. Applications to a model for nanowire growth are also provided.
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 interaction term. In the particular case $s =1$ the $s$-perimeter coincides with the classical perimeter, and our results improve the ones of Knüpfer and Muratov concerning minimality of balls of small volume in isoperimetric problems with a competition between perimeter and a nonlocal potential term. More precisely, their result is extended to its maximal range of validity concerning the type of nonlocal potentials considered, and is also generalized to the case where local perimeters are replaced by their nonlocal counterparts.
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 Mullins-Sekerka flow.
We settle the case of equality for the relative isoperimetric inequality outside any arbitrary convex set with not empty interior.
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 of an asymptotically constant forcing term we show that the only possible long time limit sets are given by disjoint unions of disks with equal radii and possibly tangent.
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.
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.
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 $Γ$-convergence we derive a two dimensional local reduced model. Investigation of this model provides an insight on the formation mechanism of {\it perpendicular magnetic anisotropy} and uniaxial anisotropy with an {\it arbitrary} preferred direction of magnetization.