We further investigate the properties of an approach to topological singularities through free discontinuity functionals of Mumford-Shah type proposed in De Luca et al. (Indiana Univ Math J 73:723–779, 2024). We prove the variational equivalence between such energies, Ginzburg-Landau, and Core-Radius for anti-plane screw dislocations energies in dimension two, in the relevant energetic regimes,, where denotes the linear size of the process zone near the defects. Further, we remove the a priori restrictive assumptions that the approximating order parameters have compact jump set. This is obtained by proving a new density result for -valued functions, approximated through functions with essentially closed jump set, in the strong BV norm.

We consider a variant of the sticky disk energy where distances between particles are evaluated through the sup norm ‖⋅‖∞in the plane. We first prove crystallization of minimizers in the square lattice, for any fixed number N of particles. Then we consider the limit as N →∞: In contrast to the standard sticky disk, there is only one orientation in the limit, and we are able to compute explicitly the Γ-limit to be an anisotropic perimeter with octagonal Wulff shape. The results are based on an energy decomposition for graphs that generalizes the one proved by De Luca-Friesecke [10] in the triangular case.

This paper deals with the dynamics-driven by the gradient flow of negative fractional seminorms-of empirical measures towards equi-spaced ground states. Specifically, we consider periodic empirical measures μ on the real line that are screened by the Lebesgue measure, i.e., with μ-d x μ-dx having zero average. To each of these measures μ we associate a (periodic) function u satisfying u ′ = d x-μ {u′=dx-μ. For s (0, 1 2) {e(0, 1{2) we introduce energy functionals s (μ) Es(μ) that can be understood as the density of the s-Gagliardo seminorm of u per unit length. Since for s ≥ 1 2 {s1/2, the s-Gagliardo seminorms are infinite on functions with jumps, some regularization procedure is needed: For s [ 1 2, 1) e1{2,1) we define ε s (μ):= s (μ ε) Es(μ):= Es(μ), where μ ε μ is obtained by mollifying μ on scale ε. We prove that the minimizers of s Es and ε s Es are the equi-spaced configurations of particles with lattice spacing equal to one. Then we prove the exponential convergence of the corresponding gradient flows to the equi-spaced steady states. Finally, although for s [ 1 2, 1) e1/2,1) the energy functionals ε s Es blow up as ε → 0 to 0, their gradients are uniformly bounded (with respect to ε), so that the corresponding trajectories converge, as ε → 0 to 0, to the gradient flow solution of a suitable renormalized energy.

In this paper we introduce the notion of parabolic α-Riesz flow, for α ∈ (0, d), extending the notion of s-fractional heat flows to negative values of the parameter s=−α2. Then, we determine the limit behaviour of these gradient flows as α → 0+ and α → 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.

We consider periodic piecewise affine functions, defined on the real line, with two given slopes, one positive and one negative, and prescribed length scale of the intervals where the slope is negative. We prove that, in such a class, the minimizers of s-fractional Gagliardo seminorm densities, with 0<1/2, are in fact periodic with the minimal possible period determined by the prescribed slopes and length scale. Then, we determine the asymptotic behavior of the energy density as the ratio between the length of the two intervals, where the slope is constant, vanishes. Our results, for s=1/2, have relevant applications to the van der Merwe theory of misfit dislocations at semicoherent straight interfaces. We consider two elastic materials having different elastic coefficients and casting parallel lattices having different spacing. As a byproduct of our analysis, we prove the periodicity of optimal dislocation configurations and we provide the sharp asymptotic energy density in the semicoherent limit as the ratio between the two lattice spacings tends to one.

We present a variational theory for lattice defects of rotational and translational type. We focus on finite systems of planar wedge disclinations, disclination dipoles, and edge dislocations, which we model as the solutions to minimum problems for isotropic elastic energies under the constraint of kinematic incompatibility. Operating under the assumption of planar linearized kinematics, we formulate the mechanical equilibrium problem in terms of the Airy stress function, for which we introduce a rigorous analytical formulation in the context of incompatible elasticity. Our main result entails the analysis of the energetic equivalence of systems of disclination dipoles and edge dislocations in the asymptotics of their singular limit regimes. By adopting the regularization approach via core radius, we show that, as the core radius vanishes, the asymptotic energy expansion for disclination dipoles coincides with the energy of finite systems of edge dislocations. This proves that Eshelby’s kinematic characterization of an edge dislocation in terms of a disclination dipole is exact also from the energetic standpoint.

We propose nonlinear semi-discrete and discrete models for the elastic energy induced by a finite system of edge dislocations in two dimensions. Within the dilute regime, we analyze the asymptotic behavior of the nonlinear elastic energy, as the core-radius (in the semi-discrete model) and the lattice spacing (in the purely discrete one) vanish. Our analysis passes through a linearization procedure within the rigorous framework of Γ-convergence.

We introduce a weak notion of $2\times 2$-minors of gradients for a suitable subclass of $BV$ functions. In the case of maps in $BV(\mathbb{R}^2; \mathbb{R}^2)$ such a notion extends the standard definition of Jacobian determinant to non-Sobolev maps. We use this distributional Jacobian to prove a compactness and $\Gamma$-convergence result for a new model describing the emergence of topological singularities in two dimensions, in the spirit of Ginzburg-Landau and core-radius approaches. Within our framework, the order parameter is an $SBV$ map $u$ taking values in the unit sphere in $\mathbb{R}^2$ and the energy is given by the sum of the squared $L^2$ norm of the approximate gradient $\nabla u$ and of the length of (the closure of) the jump set of $u$ multiplied by $\frac 1 \varepsilon$. Here, $\varepsilon$ is a length-scale parameter. We show that, in the $|\log\varepsilon|$ regime, the distributional Jacobians converge, as $\varepsilon \to 0^+$, to a finite sum $\mu$ of Dirac deltas with weights multiple of $\pi$, and that the corresponding effective energy is given by the total variation of $\mu$.

We study the asymptotic behavior, as the lattice spacing ? tends to zero, of the discrete elastic energy induced by topological singularities in an inhomogeneous ? periodic medium within a two-dimensional model for screw dislocations in the square lattice. We focus on the |log?| regime which, as ?->0 allows the emergence of a finite number of limiting topological singularities. We prove that the ?-limit of the |log?| scaled functionals as ?->0 is equal to the total variation of the so-called "limiting vorticity measure" times a factor depending on the homogenized energy density of the unscaled functionals.

This paper deals with the limit cases for s-fractional heat flows in a cylindrical domain, with homogeneous Dirichlet boundary conditions, as s-> 0+ and s-> 1-. We describe the fractional heat flows as minimizing move-ments of the corresponding Gagliardo seminorms, with re-spect to the L2 metric. To this end, we first provide a Gamma-convergence analysis for the s-Gagliardo seminorms as s-+ 0+ and s-+ 1-; then, we exploit an abstract stability result for minimizing movements in Hilbert spaces, with respect to a sequence of Gamma-converging uniformly lambda-convex energy function-als. We prove that s-fractional heat flows (suitably scaled in time) converge to the standard heat flow as s-+ 1-, and to a de-generate ODE type flow as s-+ 0+. Moreover, looking at the next order term in the asymptotic expansion of the s -fractional Gagliardo seminorm, we show that suitably forced s-fractional heat flows converge, as s-+ 0+, to the parabolic flow of an energy functional that can be seen as a sort of renormalized 0-Gagliardo seminorm: the resulting parabolic equation involves the first variation of such an energy, that can be understood as a zero (or logarithmic) Laplacian.(c) 2023 Elsevier Inc. All rights reserved.

We consider a discrete model of planar elasticity where the particles, in the reference configuration, sit on a regular triangular lattice and interact through nearest-neighbor pairwise potentials, with bonds modeled as linearized elastic springs. Within this framework, we introduce plastic slip fields, whose discrete circulation around each tri-angle detects the possible presence of an edge dislocation. We provide a gamma-convergence analysis, as the lattice spacing tends to zero, of the elastic energy induced by edge dislocations in the energy regime corresponding to a finite number of geometrically necessary dislocations.

We consider a core-radius approach to nonlocal perimeters governed by isotropic kernels having critical and supercritical exponents, extending the nowadays classical notion of s-fractional perimeter, defined for 0<1, to the case s>=1. We show that, as the core-radius vanishes, such core-radius regularized s-fractional perimeters, suitably scaled, ?-converge to the standard Euclidean perimeter. Under the same scaling, the first variation of such nonlocal perimeters gives back regularized s-fractional curvatures which, as the core radius vanishes, converge to the standard mean curvature; as a consequence, we show that the level set solutions to the corresponding nonlocal geometric flows, suitably reparametrized in time, converge to the standard mean curvature flow. Furthermore, we show the same asymptotic behavior as the core-radius vanishes and s->s?>=1 simultaneously. Finally, we prove analogous results in the case of anisotropic kernels with applications to dislocation dynamics.

We propose and analyze a class of vectorial crystallization problems, with applications to crystallization of anisotropic molecules and collective behavior such as birds flocking and fish schooling. We focus on two-dimensional systems of "oriented" particles: Admissible configurations are represented by vectorial empirical measures with density in S-1. We endow such configurations with a graph structure, where the bonds represent the "convenient" interactions between particles, and the proposed variational principle consists in maximizing their number. The class of bonds is determined by hard sphere type pairwise potentials, depending both on the distance between the particles and on the angles between the segment joining two particles and their orientations, through threshold criteria. Different ground states emerge by tuning the angular dependence in the potential, mimicking ducklings swimming in a row formation and predicting as well, for some specific values of the angular parameter, the so-called diamond formation in fish schooling.

We describe the emergence of topological singularities in periodic media within the Ginzburg-Landau model and the core-radius approach. The energy functionals of both models are denoted by E, where ? represent the coherence length (in the Ginzburg-Landau model) or the core-radius size (in the core-radius approach) and ? denotes the periodicity scale. We carry out the ? -convergence analysis of E as ?-> 0 and ?= ?-> 0 in the | log ?| scaling regime, showing that the ? -limit consists in the energy cost of finitely many vortex-like point singularities of integer degree. After introducing the scale parameter ?=min{1,lim?->0|log??||log?|}(upon extraction of subsequences), we show that in a sense we always have a separation-of-scale effect: at scales smaller than ? we first have a concentration process around some vortices whose location is subsequently optimized, while for scales larger than ? the concentration process takes place "after" homogenization.

We review some recent results on crystallization in two dimensions for pairwise interaction energies adopting a variational approach.We discuss the behavior of minimizers and quasi-minimizers of the Heitmann-Radin sticky disc model and we see how this model can be enriched in order to deal with collective behavior for systems of oriented particles.

This paper provides a unified point of view on fractional perimeters and Riesz potentials. Denoting byH? - for ? 2 .0; 1/ - the ?-fractional perimeter and by J ? - for ? 2 .(d; 0)- the ?-Riesz energies acting on characteristic functions, we prove that both functionals can be seen as limits of renormalized self-attractive energies as well as limits of repulsive interactions between a set and its complement. We also show that the functionals H? and J ? , up to a suitable additive renormalization diverging when ? ? 0, belong to a continuous one-parameter family of functionals, which for ? D 0 gives back a new object we refer to as 0-fractional perimeter. All the convergence results with respect to the parameter ? and to the renormalization procedures are obtained in the framework of A-convergence. As a byproduct of our analysis, we obtain the isoperimetric inequality for the 0-fractional perimeter.

We consider two-dimensional zero-temperature systems of N particles to which we associate an energy of the form E[V](X):=?1?iR2E[V](X)?NE ̄sq[V]+O(N12).Moreover E ̄ [V] is also re-expressed as the minimizer of a four point energy. In particular, this happens if the potential V is such that V(r) = + ? forr< 1 , V(r) = - 1 for r?[1,2], V(r) = 0 if r>2, in which case E ̄ [V] = - 4. To the best of our knowledge, this is the first proof of crystallization to the square lattice for a two-body interaction energy.

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 (Formula presented.) and (Formula presented.) In analogy with the s-fractional mean curvature flows, we introduce the notion of s-Riesz curvature flows and characterize its limit as (Formula presented.) Eventually, we discuss the limit behavior as (Formula presented.) of the flow generated by a regularization of the r-Minkowski content.

prefazione special issue

This paper deals with the variational analysis of topological singularities in two dimensions. We consider two canonical zero-temperature models: the core radius approach and the Ginzburg-Landau energy. Denoting by epsilon the length scale parameter in such models, we focus on the vertical bar log epsilon VERBAR; energy regime. It is well known that, for configurations whose energy is bounded by c vertical bar log epsilon vertical bar, the vorticity measures can be decoupled into the sum of a finite number of Dirac masses, each one of them carrying pi vertical bar log epsilon vertical bar energy, plus a mea. sure supported on small zero-average sets. Loosely speaking, on such sets the vorticity measure is close, with respect to the flat norm, to zero-average clusters of positive and negative masses. Here, we perform a compactness and Gamma-convergence analysis accounting also for the presence of such clusters of dipoles (on the range scale epsilon(s), for 0 < s < 1), which vanish in the flat convergence and whose energy contribution has, so far, been neglected. Our results refine and contain as a particular case the classical Gamma-convergence analysis for vortices, extending it also to low energy configurations consisting of just clusters of dipoles, and whose energy is of order c vertical bar log epsilon vertical bar with c < pi.