scholarly journals Equilibria of an aggregation model with linear diffusion in domains with boundaries

2020 ◽  
Vol 30 (04) ◽  
pp. 805-845
Author(s):  
Daniel Messenger ◽  
Razvan C. Fetecau
Keyword(s):  
Critical Points ◽  
Energy Minimization ◽  
Domain Boundary ◽  
Energy Functional ◽  
Linear Diffusion ◽  
Sharp Condition ◽  
Global Minimizers ◽  
Swarm Equilibria ◽  
External Forces ◽  
Domain I

We investigate the effect of linear diffusion and interactions with the domain boundary on swarm equilibria by analyzing critical points of the associated energy functional. Through this process we uncover two properties of energy minimization that depend explicitly on the spatial domain: (i) unboundedness from below of the energy due to an imbalance between diffusive and aggregative forces depends explicitly on a certain volume filling property of the domain, and (ii) metastable mass translation occurs in domains without sufficient symmetry. From the first property, we present a sharp condition for existence (respectively non-existence) of global minimizers in a large class of domains, analogous to results in free space, and from the second property, we identify that external forces are necessary to confine the swarm and grant existence of global minimizers in general domains. We also introduce a numerical method for computing critical points of the energy and give examples to motivate further research.

scholarly journals Existence of ground states for aggregation-diffusion equations

2019 ◽  
Vol 17 (03) ◽  
pp. 393-423 ◽  
Author(s):  
J. A. Carrillo ◽  
M. G. Delgadino ◽  
F. S. Patacchini
Keyword(s):  
Free Energy ◽  
Ground States ◽  
Energy Functional ◽  
Multi Agent Systems ◽  
Linear Diffusion ◽  
Sharp Condition ◽  
Macroscopic Models ◽  
Energy Functionals ◽  
Continuous Description ◽  
Degenerate Diffusions

We analyze free energy functionals for macroscopic models of multi-agent systems interacting via pairwise attractive forces and localized repulsion. The repulsion at the level of the continuous description is modeled by pressure-related terms in the functional making it energetically favorable to spread, while the attraction is modeled through nonlocal forces. We give conditions on general entropies and interaction potentials for which neither ground states nor local minimizers exist. We show that these results are sharp for homogeneous functionals with entropies leading to degenerate diffusions while they are not sharp for fast diffusions. The particular relevant case of linear diffusion is totally clarified giving a sharp condition on the interaction potential under which the corresponding free energy functional has ground states or not.

scholarly journals Regularity of the m-symphonic map

2021 ◽  
Vol 2 (2) ◽  
Author(s):  
Masashi Misawa ◽  
Nobumitsu Nakauchi
Keyword(s):  
Critical Points ◽  
Conformal Invariance ◽  
Hölder Continuity ◽  
Energy Functional ◽  
Higher Dimension ◽  
Holder Continuity ◽  
New Energy ◽  
Symphonic Map

AbstractWe introduce a new energy functional of conformal invariance and consider its critical points, named the m-symphonic map. We study a Hölder continuity of m-symphonic maps from domains of $$\mathbb {R}^m$$ R m into the spheres in the higher dimension $$m \ge 4$$ m ≥ 4 .

Harmonicity of vector fields on a class of Lorentzian solvable Lie groups

2018 ◽  
Vol 18 (3) ◽  
pp. 337-344 ◽  
Author(s):  
Ju Tan ◽  
Shaoqiang Deng
Keyword(s):  
Vector Field ◽  
Lie Groups ◽  
Lie Algebras ◽  
Critical Points ◽  
Vector Fields ◽  
Energy Functional ◽  
Smooth Vector ◽  
Solvable Lie Groups ◽  
Invariant Vector ◽  
Invariant Vector Fields

AbstractIn this paper, we consider a special class of solvable Lie groups such that for any x, y in their Lie algebras, [x, y] is a linear combination of x and y. We investigate the harmonicity properties of invariant vector fields of this kind of Lorentzian Lie groups. It is shown that any invariant unit time-like vector field is spatially harmonic. Moreover, we determine all vector fields which are critical points of the energy functional restricted to the space of smooth vector fields.

scholarly journals Harmonic maps with torsion

2020 ◽  
Author(s):  
Volker Branding
Keyword(s):  
Riemannian Manifolds ◽  
Critical Points ◽  
Mathematical Analysis ◽  
Harmonic Maps ◽  
Natural Extension ◽  
Energy Functional ◽  
Target Manifold

AbstractIn this article we introduce a natural extension of the well-studied equation for harmonic maps between Riemannian manifolds by assuming that the target manifold is equipped with a connection that is metric but has non-vanishing torsion. Such connections have already been classified in the work of Cartan (1924). The maps under consideration do not arise as critical points of an energy functional leading to interesting mathematical challenges. We will perform a first mathematical analysis of these maps which we will call harmonic maps with torsion.

scholarly journals Solutions of the multiconfiguration Dirac–Fock equations

2014 ◽  
Vol 26 (07) ◽  
pp. 1450014 ◽  
Author(s):  
Antoine Levitt
Keyword(s):  
Critical Points ◽  
Existence Of Solutions ◽  
Molecular System ◽  
Body Wave ◽  
Coupled System ◽  
Energy Functional ◽  
Relativistic Model ◽  
Occupation Numbers ◽  
Relativistic Regime ◽  
Single Configuration

The multiconfiguration Dirac–Fock (MCDF) model uses a linear combination of Slater determinants to approximate the electronic N-body wave function of a relativistic molecular system, resulting in a coupled system of nonlinear eigenvalue equations, the MCDF equations. In this paper, we prove the existence of solutions of these equations in the weakly relativistic regime. First, using a new variational principle as well as the results of Lewin on the multiconfiguration non-relativistic model, and Esteban and Séré on the single-configuration relativistic model, we prove the existence of critical points for the associated energy functional, under the constraint that the occupation numbers are not too small. Then, this constraint can be removed in the weakly relativistic regime, and we obtain non-constrained critical points, i.e. solutions of the multiconfiguration Dirac–Fock equations.

scholarly journals Solutions for a singular elliptic problem involving the p(x)-Laplacian

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 4841-4850 ◽  
Author(s):  
Khanghahi Mahdavi ◽  
A. Razani
Keyword(s):  
Bounded Domain ◽  
Critical Points ◽  
Elliptic Problem ◽  
Energy Functional ◽  
Laplacian Operator ◽  
Palais Smale Condition

Here, a singular elliptic problem involving p(x)-Laplacian operator in a bounded domain in RN is considered. Due to this, the existence of critical points for the energy functional which is unbounded below and satisfies the Palais-Smale condition are proved.

scholarly journals Towards a macroscopically consistent discrete method for granular materials: Delaunay strain-based formulation

2021 ◽  
Author(s):  
Göran Frenning
Keyword(s):  
Granular Materials ◽  
Granular Media ◽  
Deformation Gradient ◽  
Energy Functional ◽  
Computational Domain ◽  
Discrete Method ◽  
Angular Momenta ◽  
Affine Deformation ◽  
External Forces ◽  
Central Forces

AbstractWe demonstrate that the Delaunay-based strain definition proposed by Bagi (Mech Mater 22:165–177, 1996) for granular media can be straightforwardly translated into a particle-based numerical method for continua. This method has a number of attractive features, including linear completeness and satisfaction of the patch test, exact conservation of linear and angular momenta in the absence of external forces and torques, and anti-symmetry of the gradient vectors for any two points not both on the boundary of the computational domain. The formulation in effect relies on nodal (particle) interpolation of the deformation gradient and is therefore inherently unstable. Drawing on the analogy with granular media, a pairwise interaction between particles is included to alleviate this issue. The underlying idea is to define a local, non-affine deformation of each bond or contact, and to introduce pairwise forces via a stored-energy functional expressed in terms of the corresponding local displacements. In this manner, a generalisation of the Ganzenmüller (Comput Methods Appl Mech Eng 286:87–106, 2015) hourglass stabilisation procedure to non-central forces is obtained. The performance of the method is demonstrated in a range of problems. This work can be considered a first step towards the development of a macroscopically consistent discrete method for granular materials.

Harmonic sections of vector bundles with spherically symmetric metrics

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohamed T. K. Abbassi ◽  
Ibrahim Lakrini
Keyword(s):  
Riemannian Manifold ◽  
Vector Bundle ◽  
Critical Points ◽  
Vector Bundles ◽  
Energy Functional ◽  
Spherically Symmetric ◽  
Arbitrary Vector ◽  
Smooth Maps ◽  
Symmetric Metrics ◽  
Spherically Symmetric Metric

Abstract We equip an arbitrary vector bundle over a Riemannian manifold, endowed with a fiber metric and a compatible connection, with a spherically symmetric metric (cf. [4]), and westudy harmonicity of its sections firstly as smooth maps and then as critical points of the energy functional with variations through smooth sections.We also characterize vertically harmonic sections. Finally, we give some examples of special vector bundles, recovering in some situations some classical harmonicity results.

scholarly journals Solitary waves for nonlinear Klein–Gordon–Maxwell and Schrödinger–Maxwell equations

2004 ◽  
Vol 134 (5) ◽  
pp. 893-906 ◽  
Author(s):  
Teresa D'Aprile ◽  
Dimitri Mugnai
Keyword(s):  
Solitary Waves ◽  
Critical Points ◽  
Variational Approach ◽  
Maxwell Equations ◽  
Energy Functional ◽  
Nonlinear Schrödinger Equations ◽  
Mountain Pass ◽  
Schrödinger Equations ◽  
Nonlinear Schrodinger Equations ◽  
Klein Gordon

In this paper we study the existence of radially symmetric solitary waves for nonlinear Klein–Gordon equations and nonlinear Schrödinger equations coupled with Maxwell equations. The method relies on a variational approach and the solutions are obtained as mountain-pass critical points for the associated energy functional.

General Elasticity Theory for Graphene Membranes Based on Molecular Dynamics

2007 ◽  
Vol 1057 ◽  
Author(s):  
Kaveh Samadikhah ◽  
Juan Atalaya ◽  
Caroline Huldt ◽  
Andreas Isacsson ◽  
Jari Kinaret
Keyword(s):  
Molecular Dynamics ◽  
Elasticity Theory ◽  
Elastic Energy ◽  
Poisson Ratio ◽  
Md Simulations ◽  
Energy Functional ◽  
Linear Description ◽  
Continuum Description ◽  
External Forces ◽  
Graphene Membranes

ABSTRACTWe have studied the mechanical properties of suspended graphene membranes using molecular dynamics (MD) and generalized continuum elasticity theory (GE) in order to develop and assess a continuum description for graphene. The MD simulations are based on a valence force field model which is used to determine the deformation and the elastic energy of the membrane (EMD) as a function of external forces. For the continuum description, we use the expression Econt = Estretching + Ebending for the elastic energy functional. The elastic parameters (tensile rigidity and Poisson ratio) entering Econt are determined by requiring that Econt = EMD for a set of deformations.Comparisons with the MD results show excellent agreement. We find that the elastic energy of a supported graphene sheets is typically dominated by the nonlinear stretching terms whereas a linear description is valid only for very small deflections. This implies that in some applications, i.e. NEMS, a linear description is of limited applicability.

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