scholarly journals Asymptotic Behavior of Green Functions of Divergence form Operators with Periodic Coefficients

2012 ◽  
Author(s):  
X. Blanc ◽  
F. Legoll ◽  
A. Anantharaman
Keyword(s):  
Asymptotic Behavior ◽  
Divergence Form ◽  
Green Functions ◽  
Periodic Coefficients

The polynomial growth solutions to some sub-elliptic equations on the Heisenberg group

2019 ◽  
Vol 21 (01) ◽  
pp. 1750069 ◽  
Author(s):  
Hairong Liu ◽  
Tian Long ◽  
Xiaoping Yang
Keyword(s):  
Dirichlet Problem ◽  
Heisenberg Group ◽  
Elliptic Equations ◽  
Polynomial Growth ◽  
Bounded Solution ◽  
Type Theorem ◽  
Divergence Form ◽  
Liouville Type ◽  
Periodic Coefficients ◽  
Liouville Type Theorem

We give an explicit description of polynomial growth solutions to some sub-elliptic operators of divergence form with [Formula: see text]-periodic coefficients on the Heisenberg group, where the periodicity has to be meant with respect to the Heisenberg geometry. We show that the polynomial growth solutions are necessarily polynomials with [Formula: see text]-periodic coefficients. We also prove the Liouville-type theorem for the Dirichlet problem to these sub-elliptic equations on an unbounded domain on the Heisenberg group, show that any bounded solution to the Dirichlet problem must be constant.

Homogenization and correctors for the wave equation with periodic coefficients

2014 ◽  
Vol 24 (07) ◽  
pp. 1343-1388 ◽  
Author(s):  
Juan Casado-Díaz ◽  
Julio Couce-Calvo ◽  
Faustino Maestre ◽  
José D. Martín Gómez
Keyword(s):  
Asymptotic Behavior ◽  
Wave Equation ◽  
Space Variable ◽  
Limit Problem ◽  
Almost Periodic ◽  
Wave Problem ◽  
Periodic Coefficients ◽  
Convergence Method

Using the two-scale convergence method, we study the asymptotic behavior of a wave problem in ℝN with periodic coefficients in the space variable and almost-periodic coefficients in the time one. We obtain a nonlocal corrector and show how this implies that the limit problem is nonlocal in general.

scholarly journals PROSPECTS FOR DISCOVERY OF THE τ- → π-ℓ+ℓ-ντ DECAYS

2014 ◽  
Vol 35 ◽  
pp. 1460460
Author(s):  
PABLO ROIG
Keyword(s):  
Perturbation Theory ◽  
Asymptotic Behavior ◽  
Mass Spectra ◽  
Axial Vector ◽  
Form Factors ◽  
Branching Ratios ◽  
Green Functions ◽  
Vector Resonance ◽  
Chiral Perturbation ◽  
Model Independent

We study the phenomenology of the τ- → π-ντℓ+ℓ- decays (ℓ = e, μ), predicting the respective branching ratios and di-lepton invariant-mass spectra. In addition to the model-independent (QED) contributions, we investigate the structure-dependent (SD) terms, encoding features of the hadronization of QCD currents. The relevant form factors are evaluated by supplementing Chiral Perturbation Theory with the inclusion of the lightest (axial-)vector resonance multiplet as dynamical fields. The Lagrangian couplings are fully predicted requiring the known QCD asymptotic behavior to the relevant Green functions and associated form factors in the limit of an infinite number of colours. As a consequence we predict that the τ- → π-ντe+e- decays should be discovered soon while this is not granted for the ℓ = μ case.

scholarly journals Spectra of Collective Excitations and Low-Frequency Asymptotics of Green’s Functions in Uniaxial and Biaxial Ferrimagnetics

2019 ◽  
Keyword(s):  
Asymptotic Behavior ◽  
Green Function ◽  
Wave Vector ◽  
Easy Axis ◽  
Green’S Functions ◽  
Low Frequency ◽  
Easy Plane ◽  
Collective Excitations ◽  
Green Functions ◽  
The Green Function

The paper studies the dynamic description of uniaxial and biaxial ferrimagnetics with spin s=1/2 in alternative external field. The nonlinear dynamic equations with sources are obtained, on basis on which low-frequency asymptotics of two-time Green functions in the uniaxial and biaxial cases of the ferrimagnet are obtained. Energy models are constructed that are specific functions of Casimir invariants of the algebra of Poisson brackets for magnetic degrees of freedom. On their basis, the question of the stable magnetic states has been solved for the considered systems. These equations were linearized, an explicit form of the collective excitations spectra was found, and their character was analyzed. The article studies the uniaxial case of a ferrimagnet, as well as biaxial cases of an antiferromagnet, easy-axis and easy-plane ferrimagnets. It is shown that for a uniaxial antiferromagnet the spectrum of magnetic excitations has a Goldstone character. For biaxial ferrimagnetic materials, it was found that the spectrum has either a quadratic character or a more complex dependence on the wave vector. It is shown that in the uniaxial case of an antiferromagnet the Green function of the type Gsα,sβ(k,0), Gsα,nβ(k,0) and Gsα,sβ(0,ω) have regular asymptotic behavior, and the Green function of type Gnα,nβ(k,0)≈1/k2 and Gsα,nβ(0,ω)≈1/ω, Gnα,nβ(0,ω)≈1/ω2 have a pole feature in the wave vector and frequency. Biaxial ferrimagnetic states have another type of the features of low-frequency asymptotics of the Green's functions. In the case of a ferrimagnet, the “easy-axis” of the asymptotic behavior of the Green functions Gsα,sβ(0,ω), Gsα,nβ(0,ω), Gnα,nβ(0,ω), Gsα,sβ(k,0), Gsα,nβ(k,0), Gnα,nβ(k,0) have a pole character. For the case of the “easy-plane” type ferrimagnet, the asymptotics of the Green functions Gsα,nβ(0,ω), Gnα,nβ(0,ω), Gsα,nβ(k,0), Gnα,nβ(k,0), have a pole character, and the Green function Gsα,sβ(k,ω) contains both the pole component and the regular part. A comparative analysis of the low-frequency asymptotics of Green functions shows that the nature of magnetic anisotropy significantly effects the structure of low-frequency asymptotics for uniaxial and biaxial cases of ferrimagnet. Separately, we note the non-Bogolyubov character of the Green function asymptotics for ferrimagnet with biaxial anisotropy Gnα,nβ(k,0)≈1/k4.

scholarly journals Explicit Estimates for Solutions of Mixed Elliptic Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
Luisa Consiglieri
Keyword(s):  
Potential Theory ◽  
Mixed Problem ◽  
Elliptic Problems ◽  
Discontinuous Coefficients ◽  
Divergence Form ◽  
Order Equation ◽  
Second Order ◽  
Green Functions ◽  
Mixed Problems ◽  
Quantitative Estimates

We deal with the existence of quantitative estimates for solutions of mixed problems to an elliptic second-order equation in divergence form with discontinuous coefficients. Our concern is to estimate the solutions with explicit constants, for domains in ℝn (n≥2) of class C0,1. The existence of L∞ and W1,q estimates is assured for q=2 and any q<n/(n-1) (depending on the data), whenever the coefficient is only measurable and bounded. The proof method of the quantitative L∞ estimates is based on the De Giorgi technique developed by Stampacchia. By using the potential theory, we derive W1,p estimates for different ranges of the exponent p depending on the fact that the coefficient is either Dini-continuous or only measurable and bounded. In this process, we establish new existences of Green functions on such domains. The last but not least concern is to unify (whenever possible) the proofs of the estimates to the extreme Dirichlet and Neumann cases of the mixed problem.

Sign in / Sign up

Export Citation Format

Share Document