Weak Coercivity Research Articles

The existence results and Tikhonov regularization method for generalized mixed variational inequalities in Banach spaces

This paper aims to establish the Tikhonov regularization method for generalized mixed variational inequalities in Banach spaces. For this purpose, we firstly prove a very general existence result for generalized mixed variational inequalities, provided that the mapping involved has the so-called mixed variational inequality property and satisfies a rather weak coercivity condition. Finally, we establish the Tikhonov regularization method for generalized mixed variational inequalities. Our findings extended the results for the generalized variational inequality problem (for short, GVIP(F, K)) in \(R^n\) spaces (He in Abstr Appl Anal, 2012) to the generalized mixed variational inequality problem (for short, GMVIP\((F,\phi , K)\)) in reflexive Banach spaces. On the other hand, we generalized the corresponding results for the generalized mixed variational inequality problem (for short, GMVIP\((F,\phi ,K)\)) in \(R^n\) spaces (Fu and He in J Sichuan Norm Univ (Nat Sci) 37:12–17, 2014) to reflexive Banach spaces.

Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction

We study weak solutions of the homogeneous Boltzmann equation for Maxwellian molecules with a logarithmic singularity of the collision kernel for grazing collisions. Even though in this situation the Boltzmann operator enjoys only a very weak coercivity estimate, it still leads to strong smoothing of weak solutions in accordance to the smoothing expected by an analogy with a logarithmic heat equation.

Influence of bismuth substitution on yttrium orthoferrite thin films preparation by the MOD method

Yttrium orthoferrite thin films with a thickness of about 0.4μm were prepared on glass substrates by using a metal-organic decomposition method. Our studies reveal that it is possible to reduce the crystallization temperature of the yttrium orthoferrite by the substitution of the yttrium ion with bismuth. For the samples BixY1−xFeO3 with x=0.3 and x=0.4, orthorhombic yttrium orthoferrite characteristic peaks in the X-ray diffraction spectra have been detected. The lattice constants of the Bi0.3Y0.7FeO3 film were a=5.905Å, b=7.66Å, c=5.256Å with an average grain size of about 40nm. The magnetization data indicate that the film has in-plane easy axis and weak coercivity which might be explained by a possible secondary garnet phase crystallization. Faraday rotation angle of the sample was measured to be about 0.3°/µm.

Striped domains of coarse-grained magnetite observed by X-ray photoemission electron microscopy as a source of the high remanence of granites in the Vredefort dome

The characteristics of a coarse-grained high-remanence magnetite obtained from shocked Vredefort granite were investigated by X-ray magnetic circular dichroism (XMCD) analysis and X-ray absorption spectroscopy (XAS). The study utilized a spectroscopic photoelectron low-energy electron emission microscope (SPELEEM) and was conducted in the SPring-8 large-synchrotron radiation facility. It is generally believed that the strong and stable bulk remanence of Vredefort granites is due to the presence of minerals that have been strongly magnetized by either an impact-generated magnetic field or terrestrial lightning strikes. Although coarse-grained magnetite is traditionally characterized by weak coercivity and remanence, the specimen used in the present study exhibited high coercivity and an intense remanent magnetization. The presence of hematite lamellae observed on the partially oxidized magnetite specimen indicated an array of striped domains, intensifying a remanence and coercivity. We also conducted XAS and XMCD analyses on a natural lodestone permanent magnet produced by lightning strikes; while maghemite was found to be present, no magnetic domain structures were observed. Considering that the nucleation of hematite lamellae on magnetite/maghemite grains is due to high-temperature oxidation, we attribute the intense remanent magnetization and magnetic hardening of Vredefort granites to post-impact hydrothermal activity.

Synthesis and microwave-absorbing properties of Co3Fe7@C core–shell nanostructure

Co3Fe7@C core–shell nanoparticles with high performance of microwave-absorbing properties were prepared by hydrothermal method and heat treatment. The transformation of structural, morphological and magnetic properties among the carbon-encapsulated composites, which were annealed at three different temperatures, were investigated by X-ray diffraction (XRD), Fourier transform infrared spectroscopy (FTIR), transmission electron microscopy (TEM) and vibrating sample magnetometer (VSM). XRD analysis indicated the phase composition of Co3Fe7/CoFe2O4, Fe3C/Co3Fe7 and pure Co3Fe7 at different annealing temperatures. TEM confirmed the Co3Fe7@graphite core–shell nanostructure with an average particle size of 180 nm. The saturation magnetization (M s) increased monotonically with the increase in temperature, which was attributed to the crystal growth and purity of metallic core. Co3Fe7@graphite nanoparticles exhibited the hysteretic loops of soft ferromagnetic behavior with high M s of 222.85 emu g−1, weak remanent magnetization (M r) and coercivity (H c). For Co3Fe7@graphite nanomaterial, a reflection loss exceeding −20 dB was obtained between 2.8 and 10.2 GHz, which almost covering from S-band to X-band. The maximum reflection loss is −26.8 dB at 9 GHz with 1.8 mm thickness. The excellent microwave absorption properties result from the proper electromagnetic match in core–shell nanostructure and the strong natural ferromagnetic resonance.

An Energy-Minimization Finite-Element Approach for the Frank--Oseen Model of Nematic Liquid Crystals

This paper outlines an energy-minimization finite-element approach to the computational modeling of equilibrium configurations for nematic liquid crystals under free elastic effects. The method targets minimization of the system free energy based on the Frank--Oseen free-energy model. Solutions to the intermediate discretized free elastic linearizations are shown to exist generally and are unique under certain assumptions. This requires proving continuity, coercivity, and weak coercivity for the accompanying appropriate bilinear forms within a mixed finite-element framework. Error analysis demonstrates that the method constitutes a convergent scheme. Numerical experiments are performed for problems with a range of physical parameters as well as simple and patterned boundary conditions. The resulting algorithm accurately handles heterogeneous constant coefficients and effectively resolves configurations resulting from complicated boundary conditions relevant in ongoing research.

From nonlinear to linearized elasticity via Γ-convergence: The case of multiwell energies satisfying weak coercivity conditions

Linearized elasticity models are derived, via Γ-convergence, from suitably rescaled nonlinear energies when the corresponding energy densities have a multiwell structure and satisfy a weak coercivity condition, in the sense that the typical quadratic bound from below is replaced by a weaker p bound, 1 < p < 2, away from the wells. This study is motivated by, and our results are applied to, energies arising in the modeling of nematic elastomers.

Continuous/Discontinuous Galerkin methods stabilized through transfer functions applied to the incompressible elasticity and to the Stokes problem

In this work, we present a family of new Continuous/Discontinuous mixed formulations with interpolation of equal order for velocity and pressure for the Stokes problem and incompressible elasticity. The degrees of freedom associated to the discontinuous component are eliminated and an approach with computational effort similar to that of the continuous formulation is obtained. An analysis of continuity and weak coercivity (InfSup condition) is performed and it is proved that the formulation is continuous and satisfies the InfSup condition on an appropriated norm. Error estimates, derived from theorems and the lemma presented are obtained on a similar norm to the one of the GLS method. Numerical experiments with the cavity problem and a smooth solution problem show the robustness of the new formulation.

Unification of distance inequalities for linear variational problems

In this work a unifying approach is presented that leads to bounds for the distance in natural norms between solutions belonging to different spaces, of well-posed linear variational problems with the same input data. This is done in a general hilbertian framework, and in this sense, well-known inequalities such as Cea’s or Babuska’s for coercive and non-coercive problems are extended and/or refined, as mere by-products of this unified setting. More particularly such an approach gives rise to both an improvement and a generalization to the weakly coercive case, of second Strang’s inequality for abstract coercive problems. Additionally several aspects specific to linear variational problems are the subject of a thorough analysis beforehand, which also allows for clarifications and further refinements about the concept of weak coercivity.

Holmium and titanium oxide nanolaminates by atomic layer deposition

Nanolaminate (nanomultilayer) thin films of TiO2 and Ho2O3 were grown on Si(001) substrates by atomic layer deposition at 300°C from alkoxide and β-diketonate based metal precursors and ozone. Individual layer thicknesses were 2nm for TiO2 and 4.5nm for Ho2O3. As-deposited films were smooth and X-ray amorphous. After annealing at 800°C and higher temperatures the nanolaminate structure was destroyed by solid-state reaction to form Ho2Ti2O7. The films demonstrated diamagnetic or paramagnetic behaviour in the as-deposited state. After annealing, the films possessed net magnetic moment, allowing one to record saturation magnetization and weak coercivity.

Regularized Penalty Method for General Equilibrium Problems in Banach Spaces

We consider the regularized version of the penalty method for a general equilibrium problem in a Banach space setting. We suggest weak coercivity conditions instead of (generalized) monotonicity and show that they also provide weak and strong convergence properties of the method.

Field-enhanced magnetic moment in ellipsoidal nano-hematite

Bulk hematite is a canted antiferromagnet at room temperature and displays weak magnetic coercivity above the Morin transition temperature TM ∼ 262 K. Below TM, hematite displays traditional antiferromagnetic behavior, with no net magnetic moment or magnetic hysteresis. Here, we report that ellipsoidal nanocrystals of hematite (ENH) display a significant field-enhanced magnetic moment (FEMM) upon being poled by a magnetic field. This poled moment displays a giant coercive field of nearly 6000 Oe at low temperature. Atomic resolution transmission electron microscopy indicates that the nanocrystals are single crystalline, and that the surfaces are bulk-terminated. The apical terminations include the <001> sets of planes, which are implicated in possible formation of FM-arrangements near the surface. We tentatively suggest that FEMM in ENH could also arise from uncompensated surface spins or a shell of ordered spins oriented and pinned near the surface by a magnetic field. The gradual loss of magnetic moment with increasing temperature could arise as a result of competition between surface pinning energy, and kT. The large coercive field points toward possible applications for ENH in digital magnetic recording.

Iron oxide nanotubes synthesized via template-based electrodeposition

Considerable effort has been invested in the development of synthetic methods for the preparation iron oxide nanostructures for applications in nanotechnology. While a variety of structures have been reported, only a few studies have focused on iron oxide nanotubes. Here, we present details on the synthesis and characterization of iron oxide nanotubes along with a proposed mechanism for FeOOH tube formation. The FeOOH nanotubes, fabricated via a template-based electrodeposition method, are found to exhibit a unique inner-surface. Heat treatment of these tubes under oxidizing or reducing atmospheres can produce either hematite (α-Fe2O3) or magnetite (Fe3O4) structures, respectively. Hematite nanotubes are composed of small nanoparticles less than 20 nm in diameter and the magnetization curves and FC-ZFC curves show superparamagnetic properties without the Morin transition. In the case of magnetite nanotubes, which consist of slightly larger nanoparticles, magnetization curves show ferromagnetism with weak coercivity at room temperature, while FC-ZFC curves exhibit the Verwey transition at 125 K.

On penalty methods for non monotone equilibrium problems

We consider a general equilibrium problem under weak coercivity conditions in a finite-dimensional space setting. It appears such a condition provides convergence of the general penalty method without any monotonicity assumptions. We also show that the regularized version of the penalty method enables us to further weaken the coercivity condition.

Tikhonov regularization methods for inverse variational inequalities

The purpose of this paper is to study Tikhonov regularization methods for inverse variational inequalities. A rather weak coercivity condition is given which guarantees that the solution set of regularized inverse variational inequality is nonempty and bounded. Moreover, the perturbation analysis for the solution set of regularized inverse variational inequality is established. As an application, we show that solutions of regularized inverse variational inequalities form a minimizing sequence of the D-gap function under a mild condition.

Homogenization of convex functionals which are weakly coercive and not equi-bounded from above

This paper deals with the homogenization of nonlinear convex energies defined in W01,1(Ω), for a regular bounded open set Ω of RN, the densities of which are not equi-bounded from above, and which satisfy the following weak coercivity condition: There exists q>N−1 if N>2, and q⩾1 if N=2, such that any sequence of bounded energy is compact in W01,q(Ω). Under this assumption the Γ-convergence of the functionals for the strong topology of L∞(Ω) is proved to agree with the Γ-convergence for the strong topology of L1(Ω). This leads to an integral representation of the Γ-limit in C01(Ω) thanks to a local convex density. An example based on a thin cylinder with very low and very large energy densities, which concentrates to a line shows that the loss of the weak coercivity condition can induce nonlocal effects.

On the Strong Convergence of Halpern Type Proximal Point Algorithm

The main result of this paper is to prove the strong convergence of the sequence generated by the proximal point algorithm of Halpern type to a zero of a maximal monotone operator under the suitable assumptions on the parameters and error. The results extend some of the previous results or give some different conditions for convergence of the sequence. It is also indicated that when the maximal monotone operator is the subdifferential of a convex, proper, and lower semicontinuous function, the results extend all previous results in the literature. We also prove the boundedness of the sequence generated by the algorithm with a weak coercivity condition defined in the paper and without any additional assumptions on the parameters.

Exchange bias in a Co/CoO/Co trilayer with two different ferromagnetic-antiferromagnetic interfaces

The exchange-bias properties of a Co/CoO/Co sandwich structure are examined. The Co/CoO bottom bilayer was obtained by oxidation of a Co layer and the top bilayer was formed by subsequent deposition of Co on top of CoO. The strength of the interfacial coupling is shown to be stronger at the bottom Co/CoO interface than at the top interface. A number of original properties characterize exchange bias of the top bilayer. This includes weak temperature dependence of the bias field, weak coercivity due to exchange bias, and lack of the training effect. These properties suggest that the antiferromagnetic moment configuration is frozen during magnetization reversal of the top Co layer.

The Tikhonov Regularization Method for Set‐Valued Variational Inequalities

This paper aims to establish the Tikhonov regularization theory for set‐valued variational inequalities. For this purpose, we firstly prove a very general existence result for set‐valued variational inequalities, provided that the mapping involved has the so‐called variational inequality property and satisfies a rather weak coercivity condition. The result on the Tikhonov regularization improves some known results proved for single‐valued mapping.

Optical properties of ferromagnetic ytterbium‐doped III‐nitride epilayers

AbstractWe have studied the optical and magnetic properties of ytterbium (Yb) implanted GaN epilayers grown on (0001) sapphire by different epitaxial growth methods. Samples were implanted at room temperature with Yb ions and subsequently thermally annealed at 1000 °C in N2 at atmospheric pressure. The characteristic Yb3+ ion photoluminescence (PL) spectra were observed in the spectral range between 970 nm and 1050 nm. Analysis of the PL temperature quenching and PL kinetics suggest that Yb3+ ions are involved in at least two major luminescence centers. The magnetizations versus magnetic field curves show an enhancement of magnetic order for Yb‐implanted GaN epilayer containing intrinsic impurities and structural defects. The hysteretic behavior with a weak coercivity is enhanced when the magnetic field is out‐of‐plane indicating magnetic anisotropy of the Yb‐implanted GaN epilayers (© 2011 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)