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 Existence Results for the Conformal Dirac–Einstein System

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Chiara Guidi ◽  
Ali Maalaoui ◽  
Vittorio Martino
Keyword(s):  
Perturbation Methods ◽  
Existence Of Solutions ◽  
Coupled System ◽  
Existence Results ◽  
First Variation ◽  
The First Variation

AbstractWe consider the coupled system given by the first variation of the conformal Dirac–Einstein functional. We will show existence of solutions by means of perturbation methods.

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 Periodic and asymptotically periodic fourth-order Schrödinger equations with critical and subcritical growth

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Edcarlos D. Silva ◽  
Marcos L. M. Carvalho ◽  
Claudiney Goulart
Keyword(s):  
Fourth Order ◽  
Elliptic Problem ◽  
Principal Part ◽  
Existence Of Solutions ◽  
Energy Functional ◽  
Critical Nonlinearities ◽  
Asymptotically Periodic ◽  
Whole Space ◽  
Continuous Potential ◽  
Order Elliptic Problem

<p style='text-indent:20px;'>It is established existence of solutions for subcritical and critical nonlinearities considering a fourth-order elliptic problem defined in the whole space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>. The work is devoted to study a class of potentials and nonlinearities which can be periodic or asymptotically periodic. Here we consider a general fourth-order elliptic problem where the principal part is given by <inline-formula><tex-math id="M2">\begin{document}$ \alpha \Delta^2 u + \beta \Delta u + V(x)u $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M3">\begin{document}$ \alpha &gt; 0, \beta \in \mathbb{R} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ V: \mathbb{R}^N \rightarrow \mathbb{R} $\end{document}</tex-math></inline-formula> is a continuous potential. Hence our main contribution is to consider general fourth-order elliptic problems taking into account the cases where <inline-formula><tex-math id="M5">\begin{document}$ \beta $\end{document}</tex-math></inline-formula> is negative, zero or positive. In order to do that we employ some fine estimates proving the compactness for the associated energy functional.</p>

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.

Fixed point theorems for block operator matrix and an application to a structured problem under boundary conditions of Rotenberg’s model type

2014 ◽  
Vol 64 (1) ◽  
Author(s):  
Afif Amar ◽  
Aref Jeribi ◽  
Bilel Krichen
Keyword(s):  
Boundary Conditions ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Coupled System ◽  
Operator Matrix ◽  
Model Type ◽  
Block Operator Matrix ◽  
Growing Cell ◽  
Block Operator ◽  
Abstract Boundary

AbstractIn this manuscript, we introduce and study the existence of solutions for a coupled system of differential equations under abstract boundary conditions of Rotenberg’s model type, this last arises in growing cell populations. The entries of block operator matrix associated to this system are nonlinear and act on the Banach space X p:= L p([0, 1] × [a, b]; dµ dv), where 0 ≤ a < b < ∞; 1 < p < ∞.

scholarly journals Existence of Solutions for a Coupled System of Second and Fourth Order Elliptic Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Fanglei Wang
Keyword(s):  
Fourth Order ◽  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Coupled System ◽  
Monotone Sequence ◽  
Fourth Order Elliptic Equations ◽  
Quasilinearization Technique

The generalized quasilinearization technique is applied to obtain a monotone sequence of iterates converging uniformly and quadratically to a solution of a coupled system of second and fourth order elliptic equations.

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