Subharmonicity results for the stationary solutions of isotropic energy functionals

2017 ◽  
Vol 10 (2) ◽  
pp. 169-181
Author(s):  
Aleksis Koski
Keyword(s):  
Stationary Solutions ◽  
Harmonic Mappings ◽  
Jacobian Determinant ◽  
Stored Energy ◽  
Lagrange Equations ◽  
Isotropic Energy ◽  
Energy Functions ◽  
Energy Functionals ◽  
A New Technique ◽  
Derivatives Of

AbstractWe study solutions of Euler–Lagrange equations for isotropic energy functionals, generalizing a previous result on p-harmonic mappings. We classify all stored energy functions which give rise to a first-order differential expression whose Laplacian involves no third derivatives of the stationary solution. This classification gives rise to a new technique of finding subharmonicity results for the variational equations, and we also illustrate this technique in two examples. Firstly, we prove a subharmonicity result for the Jacobian determinant in the case of weighted Dirichlet energy. Secondly, we find optimal subharmonicity results in the case of a Neohookean-type stored energy function.

scholarly journals KERNEL CONVERGENCE ESTIMATES FOR DIFFUSIONS WITH CONTINUOUS COEFFICIENTS

2011 ◽  
Vol 14 (07) ◽  
pp. 979-1004
Author(s):  
CLAUDIO ALBANESE
Keyword(s):  
Convergence Rate ◽  
Convergence Rates ◽  
Fourier Transforms ◽  
Uniform Continuity ◽  
Time Step ◽  
Uniform Bounds ◽  
Kernel Convergence ◽  
Discretization Schemes ◽  
A New Technique ◽  
Derivatives Of

Bidirectional valuation models are based on numerical methods to obtain kernels of parabolic equations. Here we address the problem of robustness of kernel calculations vis a vis floating point errors from a theoretical standpoint. We are interested in kernels of one-dimensional diffusion equations with continuous coefficients as evaluated by means of explicit discretization schemes of uniform step h > 0 in the limit as h → 0. We consider both semidiscrete triangulations with continuous time and explicit Euler schemes with time step so small that the Courant condition is satisfied. We find uniform bounds for the convergence rate as a function of the degree of smoothness. We conjecture these bounds are indeed sharp. The bounds also apply to the time derivatives of the kernel and its first two space derivatives. The proof is constructive and is based on a new technique of path conditioning for Markov chains and a renormalization group argument. We make the simplifying assumption of time-independence and use longitudinal Fourier transforms in the time direction. Convergence rates depend on the degree of smoothness and Hölder differentiability of the coefficients. We find that the fastest convergence rate is of order O(h2) and is achieved if the coefficients have a bounded second derivative. Otherwise, explicit schemes still converge for any degree of Hölder differentiability except that the convergence rate is slower. Hölder continuity itself is not strictly necessary and can be relaxed by an hypothesis of uniform continuity.

scholarly journals A polyconvex extension of the logarithmic Hencky strain energy

2019 ◽  
Vol 17 (03) ◽  
pp. 349-361
Author(s):  
Robert J. Martin ◽  
Ionel-Dumitrel Ghiba ◽  
Patrizio Neff
Keyword(s):  
Strain Energy ◽  
Deformation Gradient ◽  
Singular Values ◽  
Trace Operator ◽  
Isotropic Energy ◽  
Energy Functions ◽  
Hencky Strain ◽  
Matrix Logarithm ◽  
Energy Formula ◽  
Tensor Formula

Adapting a method introduced by Ball, Muite, Schryvers and Tirry, we construct a polyconvex isotropic energy function [Formula: see text] which is equal to the classical Hencky strain energy [Formula: see text] in a neighborhood of the identity matrix 𝟙; here, [Formula: see text] denotes the set of [Formula: see text]-matrices with positive determinant, [Formula: see text] denotes the deformation gradient, [Formula: see text] is the corresponding stretch tensor, [Formula: see text] is the principal matrix logarithm of [Formula: see text], [Formula: see text] is the trace operator, [Formula: see text] is the Frobenius matrix norm and [Formula: see text] is the deviatoric part of [Formula: see text]. The extension can also be chosen to be coercive, in which case Ball’s classical theorems for the existence of energy minimizers under appropriate boundary conditions are immediately applicable. We also generalize the approach to energy functions [Formula: see text] in the so-called Valanis–Landel form [Formula: see text] with [Formula: see text], where [Formula: see text] denote the singular values of [Formula: see text].

scholarly journals Minimal coupling in presence of non-metricity and torsion

2020 ◽  
Vol 80 (8) ◽  
Author(s):  
Adrià Delhom
Keyword(s):  
Riemannian Space ◽  
Field Equations ◽  
Lagrange Equations ◽  
Minimal Coupling ◽  
Covariant Derivatives ◽  
Derivatives Of ◽  
Matter Fields

Abstract We deal with the question of what it means to define a minimal coupling prescription in presence of torsion and/or non-metricity, carefully explaining while the naive substitution $$\partial \rightarrow \nabla $$∂→∇ introduces extra couplings between the matter fields and the connection that can be regarded as non-minimal in presence of torsion and/or non-metricity. We will also investigate whether minimal coupling prescriptions at the level of the action (MCPL) or at the level of field equations (MCPF) lead to different dynamics. To that end, we will first write the Euler–Lagrange equations for matter fields in terms of the covariant derivatives of a general non-Riemannian space, and derivate the form of the associated Noether currents and charges. Then we will see that if the minimal coupling prescriptions is applied as we discuss, for spin 0 and 1 fields the results of MCPL and MCPF are equivalent, while for spin 1/2 fields there is a difference if one applies the MCPF or the MCPL, since the former leads to charge violation.

The isotropic energy function and formation rate of short gamma-ray bursts

2021 ◽  
Vol 21 (10) ◽  
pp. 254
Author(s):  
Zhi-Ying Liu ◽  
Fu-Wen Zhang ◽  
Si-Yuan Zhu
Keyword(s):  
Energy Function ◽  
Gamma Ray ◽  
Formation Rate ◽  
Gamma Ray Bursts ◽  
Local Formation ◽  
Isotropic Energy ◽  
Energy Functions ◽  
Cosmic Evolution ◽  
First Time ◽  
The Universe

Abstract Gamma-ray bursts (GRBs) are brief, intense, gamma-ray flashes in the universe, lasting from a few milliseconds to a few thousand seconds. For short gamma-ray bursts (sGRBs) with duration less than 2 seconds, the isotropic energy (E iso) function may be more scientifically meaningful and accurately measured than the luminosity (L p) function. In this work we construct, for the first time, the isotropic energy function of sGRBs and estimate their formation rate. First, we derive the L p – E p correlation using 22 sGRBs with known redshifts and well-measured spectra and estimate the pseduo redshifts of 334 Fermi sGRBs. Then, we adopt the Lynden-Bell c − method to study isotropic energy functions and formation rate of sGRBs without any assumption. A strong evolution of isotropic energy E iso ∝ (1+z)5.79 is found, which is comparable to that between L p and z. After removing effect of the cosmic evolution, the isotropic energy function can be reasonably fitted by a broken power law, which is ϕ ( E iso , 0 ) ∝ E iso , 0 − 0.45 for dim sGRBs and ϕ ( E iso , 0 ) ∝ E iso , 0 − 1.11 for bright sGRBs, with the break energy 4.92 × 1049 erg. We obtain the local formation rate of sGRBs is about 17.43 events Gpc−3 yr−1. If assuming a beaming angle is 6° to 26°, the local formation rate including off-axis sGRBs is estimated as ρ 0,all = 155.79 – 3202.35 events Gpc−3 yr−1.

scholarly journals Large elastic deformations of isotropic materials. V. The problem of flexure

1949 ◽  
Vol 195 (1043) ◽  
pp. 463-473 ◽  
Keyword(s):  
Energy Function ◽  
Incompressible Material ◽  
Curved Surface ◽  
Curved Surfaces ◽  
Stored Energy ◽  
Elastic Deformations ◽  
Axis Ii ◽  
Strain Invariants ◽  
Deformed State ◽  
Derivatives Of

A cuboid of highly elastic incompressible material, whose stored-energy function W is a function of the strain invariants, has its edges parallel to the axes x, y and z of a rectangular Cartesian co-ordinate system. It can be bent so that: (i) every plane, initially normal to the x -axis, becomes part of the curved surface of a cylinder whose axis is the z -axis; (ii) every plane, initially normal to the y -axis, becomes a plane containing the z -axis; (iii) there is no displacement parallel to the z -axis. It is found that such a state of flexure can be maintained by the application of surface tractions only, and these are calculated explicitly in terms of the derivatives of W with respect to the strain invariants. The surface tractions are normal to the surfaces on which they act, in their deformed state. Those acting on the surfaces initially normal to the x -axis are uniform over each of these surfaces. The assumption is then made that the stored-energy function W has the form, originally suggested by Mooney (1940), for rubber, W = C 1 ( λ 2 1 + λ 2 2 + λ 2 3 -3) + C 2 ( λ 2 2 λ 2 3 + λ 2 3 λ 2 1 + λ 2 1 λ 2 2 -3), where C 1 and C 2 are physical constants for the material and λ 1 , λ 2 , λ 3 are the principal extension ratios. For this case—and therefore for the incompressible neo-Hookean material (Rivlin 1948 a, b, c ), which is obtained from this by putting C 2 = 0—it is found that the flexure can be maintained without the application of surface tractions to the curved surface, provided that 2( a 1 - a 2 ) ( r 1 r 2 ) ½ = r 2 1 - r 2 2 , where ( a 1 - a 2 ) is the initial dimension of the cuboid, parallel to the x -axis, and r 1 and r 2 are the radii of the curved surfaces. When this condition is satisfied, the system of surface tractions applied to a boundary initially normal to the y -axis is equivalent to a couple M , proportional to ( C 1 + C 2 ). It is also found that the surface tractions applied to a boundary normal to the z -axis has a resultant F 2 proportional to ( C 1 - C 2 ).

scholarly journals Characterizations of Bloch-Type Spaces of Harmonic Mappings

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Munirah Aljuaid ◽  
Flavia Colonna
Keyword(s):  
Banach Space ◽  
Analytic Functions ◽  
Bloch Space ◽  
Harmonic Mappings ◽  
Open Unit ◽  
Open Unit Disk ◽  
Space Of Analytic Functions ◽  
Type Spaces ◽  
Growth Space ◽  
Derivatives Of

We study the Banach space BHα (α>0) of the harmonic mappings h on the open unit disk D satisfying the condition supz∈D⁡(1-z2)α(hzz+hz¯z)<∞, where hz and hz¯ denote the first complex partial derivatives of h. We show that several properties that are valid for the space of analytic functions known as the α-Bloch space extend to BHα. In particular, we prove that for α>0 the mappings in BHα can be characterized in terms of a Lipschitz condition relative to the metric defined by dH,α(z,w)=sup⁡{hz-hw:h∈BHα,hBHα≤1}. When α>1, the harmonic α-Bloch space can be viewed as the harmonic growth space of order α-1, while for 0<α<1, BHα is the space of harmonic mappings that are Lipschitz of order 1-α.

scholarly journals MATHEMATICAL ANALYSIS OF NONLINEAR BONDED JOINT MODELS

2004 ◽  
Vol 14 (04) ◽  
pp. 535-556 ◽  
Author(s):  
FRANCOISE KRASUCKI ◽  
ARNAUD MÜNCH ◽  
YVES OUSSET
Keyword(s):  
Mathematical Analysis ◽  
Energy Function ◽  
Three Dimensional ◽  
Expansion Method ◽  
Limit Problem ◽  
Stored Energy ◽  
Joint Models ◽  
Numerical Application ◽  
Energy Functions ◽  
The One

Within the framework of nonlinear elasticity, we consider the problem of two adherents joined along their common surface by a thin soft adhesive. Two stored energy functions are considered: the stored energy function of Saint Venant–Kirchhoff and the stored energy function of Ciarlet–Geymonat. Using the asymptotic expansion method, the limit energy associated to each of these stored energy functions is obtained. The aim of this paper is to give a rigorous mathematical analysis of the formally derived limit problem. We show that the limit problem associated to the Saint Venant–Kirchhoff case admits at least one solution and the limit problem associated to the Ciarlet–Geymonat case admits exactly one solution. An analytical comparison in the one-dimensional case and a three-dimensional numerical application are also presented.

scholarly journals Pre-Schwarzian and Schwarzian Derivatives of Harmonic Mappings

2013 ◽  
Vol 25 (1) ◽  
pp. 64-91 ◽  
Author(s):  
Rodrigo Hernández ◽  
María J. Martín
Keyword(s):  
Harmonic Mappings ◽  
Schwarzian Derivatives ◽  
Derivatives Of
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