Matteo Novaga

Matteo Novaga
Università di Pisa | UNIPI · Department of Mathematics

Professor

About

231
Publications
17,275
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3,523
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Introduction
Matteo Novaga works at the Math Department of the University of Pisa. He's interested in Applied Mathematics, Calculus of Variations and Geometric Evolutions.
Additional affiliations
October 2008 - November 2012
University of Padova
Position
  • Professor (Associate)

Publications

Publications (231)
Article
We consider the minimization problem of the functional given by the sum of the fractional perimeter and a general Riesz potential, which is one generalization of Gamow’s liquid drop model. We first show the existence of minimizers for any volumes if the kernel of the Riesz potential decays faster than that of the fractional perimeter. We also prove...
Article
We show the existence of generalized clusters of a finite or even infinite number of sets, with minimal total perimeter and given total masses, in metric measure spaces homogeneous with respect to a group acting by measure preserving homeomorphisms, for a quite wide range of perimeter functionals. Such generalized clusters are a natural “relaxed” v...
Preprint
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The motion by curvature of networks is the generalization to finite union of curves of the curve shortening flow. This evolution has several peculiar features, mainly due to the presence of junctions where the curves meet. In this paper we show that whenever the length of one single curve vanishes and two triple junctions coalesce, then the curvatu...
Article
In this paper we discuss existence, uniqueness and some properties of a class of solitons to the anisotropic mean curvature flow, i.e., graphical translators, either in the plane or under an assumption of cylindrical symmetry on the anisotropy and the mobility. In these cases, the equation becomes an ordinary differential equation, and this allows...
Preprint
Full-text available
We consider the volume constrained fractional mean curvature flow of a nearly spherical set, and prove long time existence and asymptotic convergence to a ball. The result applies in particular to convex initial data, under the assumption of global existence. Similarly, we show exponential convergence to a constant for the fractional mean curvature...
Article
Full-text available
We study a geometric variational problem arising from modeling two-dimensional charged drops of a perfectly conducting liquid in the presence of an external potential. We characterize the semicontinuous envelope of the energy in terms of a parameter measuring the relative strength of the Coulomb interaction. As a consequence, when the potential is...
Article
Full-text available
We consider the fractional mean curvature flow of entire Lipschitz graphs. We provide regularity results, and we study the long time asymptotics of the flow. In particular we show that in a suitable rescaled framework, if the initial graph is a sublinear perturbation of a cone, the evolution asymptotically approaches an expanding self-similar solut...
Preprint
Full-text available
We consider a variational problem involving competition between surface tension and charge repulsion. We show that, as opposed to the case of weak (short-range) interactions where we proved ill-posedness of the problem in a previous paper, when the repulsion is stronger the perimeter dominates the capacitary term at small scales. In particular we p...
Preprint
Full-text available
We consider the minimization problem of the functional given by the sum of the fractional perimeter and a general Riesz potential, which is one generalization of Gamow's liquid drop model. We first show the existence of minimizers for any volumes if the kernel of the Riesz potential decays faster than that of the fractional perimeter. Secondly, we...
Preprint
Full-text available
We show the existence of generalized clusters of a finite or even infinite number of sets, with minimal total perimeter and given total masses, in metric measure spaces homogeneous with respect to a group acting by measure preserving homeomorphisms, for a quite wide range of perimeter functionals. Such generalized clusters are a natural "relaxed''...
Article
We consider the anisotropic mean curvature flow of entire Lipschitz graphs. We prove existence and uniqueness of expanding self-similar solutions which are asymptotic to a prescribed cone, and we characterize the long time behavior of solutions, after suitable rescaling, when the initial datum is a sublinear perturbation of a cone. In the case of r...
Article
We prove existence of weak solutions to the obstacle problem for semilinear wave equations (including the fractional case) by using a suitable approximating scheme in the spirit of minimizing movements. This extends the results in Bonafini et al. (2019), where the linear case was treated. In addition, we deduce some compactness properties of concen...
Article
We consider a version of Gamow’s liquid drop model with a short range attractive perimeter-penalizing potential and a long-range Coulomb interaction of a uniformly charged mass in R 3 . Here we constrain ourselves to minimizing among the class of shapes that are columnar, i.e., constant in one spatial direction. Using the standard perimeter in the...
Article
We consider sets in RN which minimise, for fixed volume, the sum of the perimeter and a non-local term given by the double integral of a kernel g:RN∖{0}→R+. We establish some general existence and regularity results for minimisers. In the two-dimensional case we show that balls are the unique minimisers in the perimeter-dominated regime, for a wide...
Preprint
Full-text available
In this paper we discuss existence, uniqueness and some properties of a class of solitons to the anisotropic mean curvature flow, i.e., graphical translators, either in the plane or under an assumption of cylindrical symmetry on the anisotropy and the mobility. In these cases, the equation becomes an ordinary differential equation, and this allows...
Preprint
Full-text available
We study the regularity of solutions to a nonlocal version of the image denoising model and we show that, in two dimensions, minimizers have the same H\"older regularity as the original image. More precisely, if the datum is (locally) $\beta$-H\"older continuous for some $\beta\in(1-s,\,1]$, where $s\in (0,1)$ is a parameter related to the nonlocal...
Preprint
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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...
Article
Full-text available
This paper establishes a theory of nonlinear spectral decompositions by considering the eigenvalue problem related to an absolutely one-homogeneous functional in an infinite-dimensional Hilbert space. This approach is both motivated by works for the total variation, where interesting results on the eigenvalue problem and the relation to the total v...
Preprint
Full-text available
We consider the anisotropic mean curvature flow of entire Lipschitz graphs. We prove existence and uniqueness of expanding self-similar solutions which are asymptotic to a prescribed cone, and we characterize the long time behavior of solutions, after suitable rescaling, when the initial datum is a sublinear perturbation of a cone. In the case of r...
Article
Full-text available
We consider motion by anisotropic curvature of a network of three curves immersed in the plane meeting at a triple junction and with the other ends fixed. We show existence, uniqueness and regularity of a maximal geometric solution and we prove that, if the maximal time is finite, then either the length of one of the curves goes to zero or the $$L^...
Preprint
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The aim of this note is to prove that quasi-minimizers of the perimeter are Reifenberg flat, for a very weak notion of quasi-minimality. The main observation is that smallness of the excess at some scale implies smallness of the excess at all smaller scales.
Preprint
Full-text available
We consider the fractional mean curvature flow of entire Lipschitz graphs. We provide regularity results, and we study the long time asymptotics of the flow. In particular we show that in a suitable rescaled framework, if the initial graph is a sublinear perturbation of a cone, the evolution asymptotically approaches an expanding self-similar solut...
Article
Full-text available
We consider an energy functional combining the square of the local oscillation of a one-dimensional function with a double-well potential. We establish the existence of minimal heteroclinic solutions connecting the two wells of the potential. This existence result cannot be accomplished by standard methods, due to the lack of compactness properties...
Article
We introduce a notion of uniform convergence for local and nonlocal curvatures. Then, we propose an abstract method to prove the convergence of the corresponding geometric flows, within the level set formulation. We apply such a general theory to characterize the limits of s-fractional mean curvature flows as s → 0 + and s → 1 − . In analogy with t...
Preprint
Full-text available
We consider motion by anisotropic curvature of a network of three curves immersed in the plane meeting at a triple junction and with the other ends fixed. We show existence, uniqueness and regularity of a maximal geometric solution and we prove that, if the maximal time is finite, then either the length of one of the curves goes to zero or the $L^2...
Preprint
Full-text available
We consider the evolution of sets by nonlocal mean curvature and we discuss the preservation along the flow of two geometric properties, which are the mean convexity and the outward minimality. The main tools in our analysis are the level set formulation and the minimizing movement scheme for the nonlocal flow. When the initial set is outward minim...
Preprint
We prove existence of weak solutions to the obstacle problem for semilinear wave equations (including the fractional case) by using a suitable approximating scheme in the spirit of minimizing movements. This extends the results in [9], where the linear case was treated. In addition, we deduce some compactness properties of concentration sets (e.g....
Article
We study the rate of convergence of some nonlocal functionals recently considered by Bourgain, Brezis, and Mironescu, and, after a suitable rescaling, we establish the Γ-convergence of the corresponding rate functionals to a limit functional of second order.
Article
We prove existence of partitions of an open set Ω with a given number of phases, which minimize the sum of the fractional perimeters of all the phases, with Dirichlet boundary conditions. In two dimensions we show that, if the fractional parameter s is sufficiently close to 1, the only singular minimal cone, that is, the only minimal partition inva...
Chapter
In a previous work by the authors a second order gradient flow of the p-elastic energy for a planar theta-network of three curves with fixed lengths was considered and a weak solution of the flow was constructed by means of an implicit variational scheme. Long-time existence of the evolution and convergence to a critical point of the energy were sh...
Preprint
Full-text available
We consider sets in $\mathbb R^N$ which minimise, for fixed volume, the sum of the perimeter and a non-local term given by the double integral of a kernel $g:\mathbb R^N\setminus\{0\}\to \mathbb R^+$. We establish some general existence and regularity results for minimisers. In the two-dimensional case we show that balls are the unique minimisers i...
Preprint
Full-text available
We consider a version of Gamow's liquid drop model with a short range attractive perimeter-penalizing potential and a long-range Coulomb interaction of a uniformly charged mass in $\R^3$. Here we constrain ourselves to minimizing among the class of shapes that are columnar, i.e., constant in one spatial direction. Using the standard perimeter in th...
Preprint
We study a geometric variational problem arising from modeling two-dimensional charged drops of a perfectly conducting liquid in the presence of an external potential. We characterize the semicontinuous envelope of the energy in terms of a parameter measuring the relative strength of the Coulomb interaction. As a consequence, when the potential is...
Article
Full-text available
We introduce a notion of connected perimeter for planar sets defined as the lower semicontinuous envelope of perimeters of approximating sets which are measure-theoretically connected. A companion notion of simply connected perimeter is also studied. We prove a representation formula which links the connected perimeter, the classical perimeter, and...
Article
For a given family of smooth closed curves $γ^1,...,γ^\alpha⊂\mathbb{R}^3$ we consider the problem of finding an elastic connected compact surface $M$ with boundary $γ=γ^1\cup...\cupγ^\alpha$. This is realized by minimizing the Willmore energy $\mathcal{W}$ on a suitable class of competitors. While the direct minimization of the Area functional may...
Preprint
Full-text available
We consider a variational scheme for the anisotropic (including crystalline) mean curvature flow of sets with strictly positive anisotropic mean curvature. We show that such condition is preserved by the scheme, and we prove the strict convergence in BV of the time-integrated perimeters of the approximating evolutions, extending a recent result of...
Preprint
Full-text available
We introduce a notion of uniform convergence for local and nonlocal curvatures. Then, we propose an abstract method to prove the convergence of the corresponding geometric flows, within the level set formulation. We apply such a general theory to characterize the limits of $s$-fractional mean curvature flows as $s\to 0^+$ and $s\to 1^-$. In analogy...
Preprint
Full-text available
In this note we provide a second-order asymptotic expansion of the fractional perimeter Per$_s(E)$, as $s\to 1^-$, in terms of the local perimeter and of a higher order nonlocal functional.
Article
We consider an obstacle problem for (possibly non-local) wave equations, and we prove existence of weak solutions through a convex minimization approach based on a time discrete approximation scheme. We provide the corresponding numerical implementation and raise some open questions.
Preprint
Full-text available
We show existence of nonlocal minimal cluster with Dirichlet boundary data. In two dimensions we show that, if the fractional parameter s is sufficiently close to 1, the only singular minimal cone consists of three half-lines meeting at 120 degrees at a common end-point. In the case of fractional perimeter with weights, we show that there exists a...
Preprint
Full-text available
For a given family of smooth closed curves $\gamma^1,...,\gamma^\alpha\subset\mathbb{R}^3$ we consider the problem of finding an elastic \emph{connected} compact surface $M$ with boundary $\gamma=\gamma^1\cup...\cup\gamma^\alpha$. This is realized by minimizing the Willmore energy $\mathcal{W}$ on a suitable class of competitors. While the direct m...
Article
Full-text available
We discuss fattening phenomenon for the evolution of sets according to their nonlocal curvature. More precisely, we consider a class of generalized curvatures which correspond to the first variation of suitable nonlocal perimeter functionals, defined in terms of an interaction kernel K, which is symmetric, nonnegative, possibly singular at the orig...
Chapter
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We present a collection of results on the evolution by curvature of networks of planar curves. We discuss in particular the existence of a solution and the analysis of singularities.
Preprint
Full-text available
We introduce a notion of connected perimeter for planar sets defined as the lower semi-continuous envelope of perimeters of approximating sets which are measure-theoretically connected. A companion notion of simply connected perimeter is also studied. We prove a representation formula which links the connected perimeter, the classical perimeter, an...
Preprint
Full-text available
We study the rate of convergence of some nonlocal functionals recently considered by Bourgain, Brezis, and Mironescu. In particular, we establish the Γ-convergence of the corresponding rate functionals, suitably rescaled, to a limit functional of second order.
Preprint
Full-text available
This paper provides a unified point of view on fractional perimeters and Riesz potentials. Denoting by $H^\sigma$ - for $\sigma\in (0,1)$ - the $\sigma$-fractional perimeter and by $J^\sigma$ - for $\sigma\in (-d,0)$ - the $\sigma$-Riesz energies acting on characteristic functions, we prove that both functionals can be seen as limits of renormalize...
Article
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We show existence of homothetically shrinking solutions of the fractional mean curvature flow, whose boundary consists in a prescribed number of concentric spheres. We prove that all these solutions, except from the ball, are dynamically unstable.
Preprint
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We present a collection of results on the evolution by curvature of networks of planar curves. We discuss in particular the existence of a solution and the analysis of singularities.
Preprint
Full-text available
We consider a second order gradient flow of the p-elastic energy for a planar theta-network of three curves with fixed lengths. We construct a weak solution of the flow by means of an implicit variational scheme. We show long-time existence of the evolution and convergence to a critical point of the energy.
Preprint
Full-text available
We consider obstacle type problems for (non-local) wave equations, modelled by the (fractional) Laplacian, and we prove existence of weak solutions through a convex minimization approach based on a time discrete approximation scheme. We provide the corresponding numerical implementation and state some open questions related to the problem.
Preprint
Full-text available
The aim of this paper is to establish a theory of nonlinear spectral decompositions in an infinite dimensional setting by considering the eigenvalue problem related to an absolutely one-homogeneous functional in a Hilbert space. This approach is motivated by works for the total variation and related functionals in L 2 , where some interesting resul...
Article
Full-text available
We consider a potential W: ℝ m → ℝ with two different global minima a - , a + and, under a symmetry assumption, we use a variational approach to show that the Hamiltonian system ü = W u (u), has a family of T-periodic solutions u T which, along a sequence T j → +∞, converges locally to a heteroclinic solution that connects a - to a + . We then focu...
Preprint
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We show existence of homothetically shrinking solutions of the fractional mean curvature flow, whose boundary consists in a prescribed numbers of concentric spheres. We prove that all these solutions, except from the ball, are dynamically unstable.
Article
We consider low-energy configurations for the Heitmann–Radin sticky discs functional, in the limit of diverging number of discs. More precisely, we renormalize the Heitmann–Radin potential by subtracting the minimal energy per particle, i.e. the so-called kissing number. For configurations whose energy scales like the perimeter, we prove a compactn...
Preprint
Full-text available
We consider an energy functional combining the square of the local oscillation of a one--dimensional function with a double well potential. We establish the existence of minimal heteroclinic solutions connecting the two wells of the potential. This existence result cannot be accomplished by standard methods, due to the lack of compactness propertie...
Preprint
Full-text available
This paper is concerned with the diffuse interface Ohta-Kawasaki energy in three space dimensions, in a periodic setting, in the parameter regime corresponding to the onset of non-trivial minimizers. We identify the scaling in which a sharp transition from asymptotically trivial to non-trivial minimizers takes place as the small parameter character...
Preprint
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We develop a phase-field approximation of the relaxation of the perimeter functional in the plane under a connectedness constraint based on the classical Modica-Mortola functional and the connectedness constraint of (Dondl, Lemenant, Wojtowytsch 2017). We prove convergence of the approximating energies and present numerical results and applications...
Article
Full-text available
We consider a planar geometric flow in which the normal velocity is a nonlocal variant of the curvature. The flow is not scaling invariant and in fact has different behaviors at different spatial scales, thus producing phenomena that are different with respect to both the classical mean curvature flow and the fractional mean curvature flow. In part...
Preprint
Full-text available
We discuss fattening phenomenon for the evolution of planar curves according to their nonlocal curvature. More precisely, we consider a class of generalized curvatures which correspond to the first variation of suitable nonlocal perimeter functionals, defined in terms of an interaction kernel $K$, which is symmetric, nonnegative, possibly singular...
Preprint
Full-text available
We consider a potential $W:R^m\rightarrow R$ with two different global minima $a_-, a_+$ and, under a symmetry assumption, we use a variational approach to show that the Hamiltonian system \begin{equation} \ddot{u}=W_u(u), \hskip 2cm (1) \end{equation} has a family of $T$-periodic solutions $u^T$ which, along a sequence $T_j\rightarrow+\infty$, con...
Preprint
Full-text available
We consider low energy configurations for the Heitmann-Radin sticky discs functional, in the limit of diverging number of discs. More precisely, we renormalize the Heitmann-Radin potential by subtracting the minimal energy per particle, i.e., the so called kissing number. For configurations whose energy scales like the perimeter, we prove a compact...
Article
Full-text available
We study a nonlocal perimeter functional inspired by the Minkowski content, whose main feature is that it interpolates between the classical perimeter and the volume functional. This problem is related by a generalized coarea formula to a Dirichlet energy functional in which the energy density is the local oscillation of a function. These two nonlo...
Article
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We define a family of functionals, called p-oscillation functionals, that can be interpreted as discrete versions of the classical total variation functional for p=1 and of the p-Dirichlet functionals for p>1. We introduce the notion of minimizers and prove existence of solutions to the Dirichlet problem. Finally we provide a description of Class A...
Article
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We study the localization of sets with constant nonlocal mean curvature and prescribed small volume in a bounded open set with smooth boundary, proving that they are {\em sufficiently close} to critical points of a suitable non-local potential. We then consider the fractional perimeter in half-spaces. We prove the existence of a minimizer under fix...
Article
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We discuss a variational model, given by a weighted sum of perimeter, bending and Riesz interaction energies, that could be considered as a toy model for charged elastic drops. The different contributions have competing preferences for strongly localized and maximally dispersed structures. We investigate the energy landscape in dependence of the si...
Article
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We consider planar networks of three curves that meet at two junctions with prescribed equal angles, minimizing a combination of the elastic energy and the length functional. We prove existence and regularity of minimizers, and we show some properties of the minimal configurations.
Article
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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.
Article
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Assume that \(W:\mathbb {R}^m\rightarrow \mathbb {R}\) is a nonnegative potential that vanishes only on a finite set A with at least two elements. By direct minimization of the action functional on a suitable set of maps we give a new elementary proof of the existence of a heteroclinic orbit that connects any given \(a_-\in A\) to some \(a_{+}\in A...
Article
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We describe the macroscopic behavior of evolutions by crystalline curvature of planar sets in a chessboard--like medium, modeled by a periodic forcing term. We show that the underlying microstructure may produce both pinning and confinement effects on the geometric motion.
Article
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We consider a planar geometric flow in which the normal velocity is a nonlocal variant of the curvature. The flow is not scaling invariant and in fact has different behaviors at different spatial scales, thus producing phenomena that are different with respect to both the classical mean curvature flow and the fractional mean curvature flow. In part...
Article
Full-text available
In this paper we raise the question whether every closed Riemannian manifold has a spine of minimal area, and we answer it affirmatively in the surface case. On constant curvature surfaces we introduce the spine systole, a continuous real function on moduli space that measures the minimal length of a spine in each surface. We show that the spine sy...
Article
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We prove a quantitative version of the isoperimetric inequality for a non local perimeter of Minkowski type. We also apply this result to study isoperimetric problems with repulsive interaction terms, under convexity constraints. We show existence of minimizers, and we describe the shape of minimizers in certain parameter regimes.
Article
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We consider a class of nonlocal generalized perimeters which includes fractional perimeters and Riesz type potentials. We prove a general isoperimetric inequality for such functionals, and we discuss some applications. In particular we prove existence of an isoperimetric profile, under suitable assumptions on the interaction kernel.
Article
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We consider the evolution by crystalline curvature of a planar set in a stratified medium, modeled by a periodic forcing term. We characterize the limit evolution law as the period of the oscillations tends to zero. Even if the model is very simple, the limit evolution problem is quite rich, and we discuss some properties such as uniqueness, compar...