Exact Growth Condition Research Articles

Sharp Higher Order Adams’ Inequality with Exact Growth Condition on Weighted Sobolev Spaces

Sharp Higher Order Adams’ Inequality with Exact Growth Condition on Weighted Sobolev Spaces

The Role of Oxygen Transfer in Oxide Heterostructures on Functional Properties

AbstractA variety of mechanisms are reported to play critical roles in contributing to the high carrier/electron mobility in oxide/SrTiO3 (STO) heterostructures. By using La0.95Sr0.05TiO3 (LSTO) epitaxially grown on different single crystal substrates (such as STO, GdScO3, LaAlO3, (LaAlO3)0.3(Sr2AlTaO6)0.7, and CeO2 buffered STO) as the model systems, the formation of a conducting substrate surface layer (CSSL) on STO substrate is shown at relatively low growth temperature and high oxygen pressure (725 °C, 5 × 10–4 Torr), which contributes to the enhanced conductivity of the LSTO/STO heterostructures. Different from the conventional oxygen vacancy model, this work reveals that the formation of the CSSL occurs when growing an oxide layer (LSTO in this case) on STO, while neither annealing nor the growth of an Au layer alone at the exact same growth condition generates the CSSL in STO. It demonstrates that the oxide layer actively pulls oxygen from STO substrate at given growth conditions, leading to the formation of the CSSL. The observations emphasize the oxygen transfer across film/substrate interface during the synthesis of oxide heterostructures playing a critical role in functional properties.

Adams inequalities with exact growth condition for Riesz-like potentials on [formula omitted

We derive sharp Adams inequalities with exact growth condition for the Riesz potential and for more general Riesz-like potentials on Rn. We also obtain Moser-Trudinger inequalities with exact growth condition for the fractional Laplacian, and for general homogeneous elliptic differential operators with constant coefficients.

Sharp Adams–Moser–Trudinger type inequalities in the hyperbolic space

The purpose of this paper is to establish some Adams–Moser–Trudinger inequalities, which are the borderline cases of the Sobolev embedding, in the hyperbolic space \mathbb H^n . First, we prove a sharp Adams’ inequality of order two with the exact growth condition in \mathbb H^n . Then we use it to derive a sharp Adams-type inequality and an Adachi–Tanakat-ype inequality. We also prove a sharp Adams-type inequality with Navier boundary condition on any bounded domain of \mathbb H^n , which generalizes the result of Tarsi to the setting of hyperbolic spaces. Finally, we establish a Lions-type lemma and an improved Adams-type inequality in the spirit of Lions in \mathbb H^n . Our proofs rely on the symmetrization method extended to hyperbolic spaces.

Higher order Adams’ inequality with the exact growth condition

Adams’ inequality is the complete generalization of the Trudinger–Moser inequality to the case of Sobolev spaces involving higher order derivatives. The failure of the original form of the sharp inequality when the problem is considered on the whole space [Formula: see text] served as a motivation to investigate in the direction of a refined sharp inequality, the so-called Adams’ inequality with the exact growth condition. Due to the difficulties arising in the higher order case from the lack of direct symmetrization techniques, this refined result is known to hold on first- and second-order Sobolev spaces only. We extend the validity of Adams’ inequality with the exact growth to higher order Sobolev spaces.

Adams inequality with exact growth in the hyperbolic space ℍ4 and Lions lemma

In this paper, we prove Adams inequality with exact growth condition in the four-dimensional hyperbolic space [Formula: see text] [Formula: see text] [Formula: see text]. We will also establish an Adachi–Tanaka-type inequality in this setting. Another aspect of this paper is the Lions lemma in the hyperbolic space. We prove Lions lemma for the Moser functional and for a few cases of the Adams functional on the whole hyperbolic space.

Influence of HVPE substrates on homoepitaxy of GaN grown by MOCVD

The availability of high quality, free-standing GaN substrates enables new device applications in III-nitrides, especially for vertical device structures. With the introduction of these native substrates, the properties of nitrides are no longer dominated by defects introduced by heteroepitaxial growth. However, additional materials challenges are coming to the forefront that need to be understood and surmounted in order to allow homoepitaxial devices to achieve their full potential. In this paper, 2 μm of UID GaN are grown simultaneously by MOCVD on three commercially sourced 2″ HVPE wafers. By doing so, the substrates are exposed to the exact same growth conditions and the influence of the substrate can be unambiguously identified. The results are presented in regards to the effects of the substrates on epitaxial film morphology, uniformity, impurity incorporation, substrate/epitaxy interface, and electrical properties.

Compositional segregation and electrical properties characterization of [001]- and [011]-oriented co-growth Pb(In1/2Nb1/2)O3-Pb(Mg1/3Nb2/3)O3-PbTiO3 single crystal

A Pb(In1/2Nb1/2)O3-Pb(Mg1/3Nb2/3)O3-PbTiO3 relaxor ferroelectric single crystal boule of 3 in. diameter was grown using [001]- and [011]-oriented co-growth crystals as seeds by the modified Bridgman method. The as-grown crystal boule was divided into two growth regions [001] and [011]. The composition and the electrical properties of samples of the two regions were characterized in detail along the growth direction in the whole crystal boule. The PT content of [001]-grown samples increased from 27.7 to 36.8 mol. % along the growth direction, especially they are averagely higher about 0.7 mol. % than those of [011]-grown samples at the same growth position. The PMN content of [001]-grown samples decreased from 43.0 to 34.8 mol. %, which is lower than that of the corresponding [011]-grown samples. The compositional segregation behavior along [001] and [011] growth directions was discussed based on the crystal growth theories. Variations in the rhombohedral to tetragonal phase transition temperature (TRT), the piezoelectric properties, the coercive electric field and the remnant polarization of [001] samples were discussed according to changes in compositional segregation and crystal anisotropy. Under the exact same growth conditions, the [011]-grown crystal has a larger composition segregation effect than that of the [001]-grown crystal. This result provides a positive effect on analysing the nature and reducing the effect of compositional segregation.

Singular Moser-Trudinger inequality with the exact growth condition on hyperbolic space

In this paper, we are concerned with a singular version of the Moser-Trudinger inequality with the exact growth condition in the n-dimension hyperbolic space mathbb{H}^{n}. Our result is a natural extension of the work of Lu and Tang in (J. Geom. Anal. 26:837-857, 2016).

Singular Moser-Trudinger inequality with the exact growth condition in ℝ^n

Singular Moser-Trudinger inequality with the exact growth condition in ℝ^n

Singular Moser-Trudinger inequality with the exact growth condition in ℝ^n

Singular Moser-Trudinger inequality with the exact growth condition in ℝ^n

Best Constants for Adams’ Inequalities with the Exact Growth Condition in ℝn

Abstract In this paper, we establish the following sharp Adams inequality with exact growth condition in the entire space ℝn (n ≥ 3): There exists a constant C(n) > 0 such that for all with , , where . This extends the main result in [27] when n = 4 to all dimensions n ≥ 3. A crucial technical lemma we need is Lemma 4.2 for all p > 1 (corresponding to the Adams inequality for all n ≥ 3) whose proof is quite involved. As an application, we obtain the best constant for Ozawa’s inequality of Adams type in the Sobolev space : For any α < βn, there exists a constant C(α, n) > 0 such that for all satisfying , we have . Moreover, if α ≥ βn then the inequality cannot hold with a uniform constant C(α, n).

Trudinger-Moser Inequalities with the Exact Growth Condition in ℝ N and Applications

In a recent paper [19], the authors obtained a sharp version of the Trudinger-Moser inequality in the whole space ℝ2, giving necessary and sufficient conditions for the boundedness and the compactness of general nonlinear functionals in W 1, 2(ℝ2). We complete this study showing that an analogue of the result in [19] holds in arbitrary dimensions N ≥2. We also provide an application to the study of the existence of ground state solutions for quasilinear elliptic equations in ℝ N .

Trudinger–Moser inequality on the whole plane with the exact growth condition

Trudinger-Moser inequality is a substitute to the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to L^{\infty} . It is well known that the original form of the inequality with the sharp exponent (proved by Moser) fails on the whole plane, but a few modified versions are available. We prove a precised version of the latter, giving necessary and sufficient conditions for the boundedness, as well as for the compactness, in terms of the growth and decay of the nonlinear function. It is tightly related to the ground state of the nonlinear Schrödinger equation (or the nonlinear Klein-Gordon equation), for which the range of the time phase (or the mass constant) as well as the energy is given by the best constant of the inequality.

Sharp Moser–Trudinger Inequalities on Hyperbolic Spaces with Exact Growth Condition

Let \(\Phi _{n}(x)=e^x-\sum _{j=0}^{n-2}\frac{x^j}{j!}\) and \(\alpha _{n} =n\omega _{n-1}^{\frac{1}{n-1}}\) be the sharp constant in Moser’s inequality (where \(\omega _{n-1}\) is the area of the surface of the unit \(n\)-ball in \(\mathbb {R}^n\)), and \(dV\) be the volume element on the \(n\)-dimensional hyperbolic space \((\mathbb {H}^n, g)\) (\(n\ge {2}\)). In this paper, we establish the following sharp Moser–Trudinger type inequalities with the exact growth condition on \(\mathbb {H}^n\):

Adams' Inequality with the Exact Growth Condition in ℝ4

Adams' inequality is an extension of the Trudinger‐Moser inequality to the case when the Sobolev space considered has more than one derivative. The goal of this paper is to give the optimal growth rate of the exponential‐type function in Adams' inequality when the problem is considered in the whole space ℝ4. © 2014 Wiley Periodicals, Inc.

Intrinsic Polarised Emission from InAs/GaAs(311)A Quantum Dots

Dependence on the exact growth condition of the InAs/GaAs quantum dot emission polarisation has been tested for (100) and (311)A substrates. Polarised emission, reproducible in intensity and direction and independent of the coverage, has been achieved by self-assembling InAs quantum dots on (311)A GaAs substrates only. The quantum dots emission is stably polarised along the [233] crystallographic direction with a polarisation ratio of about 12%.

Material properties of InAlAs layers grown by MBE on vicinal (1 1 1)B InP substrates

InAlAs layers and InGaAs/InAlAs quantum well heterostructures, grown by molecular beam epitaxy on vicinal (1 1 1)B InP substrates, exhibited extensive surface step bunching, compositional inhomogeneities and degraded crystalline quality, with variations depending on the exact misfit strain and growth conditions. Silicon and beryllium doping of (1 1 1) InAlAs behaved similarly to the (1 0 0) case but a different predominant deep level was detected in n-type (1 1 1) material.

NMR studies of 59Co in Cu/Co/Cu(111) trilayer systems grown by MBE

NMR using the 59Co nucleus has been measured for a series of Cu/Co/Cu(111) trilayers grown by MBE with Co thicknesses ranging from 25 to 178 Å. The Co layer retains the Cu fcc structure up to a thickness of ∼ 60 Å at which point the hcp structure is favoured. This structural transition is not identical for each sample suggesting that the point at which the transition occurs depends on the exact growth conditions including the thickness of the Cu underlayer. Our results suggest that for very thin (15 Å) Cu underlayers the transition to hcp occurs at thicknesses less than 60 Å.

TEM study of very thin InGaAsP layers grown by liquid-phase epitaxy

The very thin (<100 Å) InGaAsP layers were grown not only by molecular beam epitaxy and metal-organic chemical vapor deposition but recently also by simple liquid phase epitaxy (LPE) technique. Characterization of their thickness, interfase abruptness and lattice defects is important and requires TEM methods to be used.The samples were InGaAsP/InGaP double heterostructures grown on (111)A GaAs substrate. The exact growth conditions are described in Ref.1. The salient points are that the quarternary layers were being grown at 750°C during a fast movement of substrate and a convection caused in the melt by that movement was eliminated. TEM cross-section specimens were prepared by means of conventional procedure. The studies were conducted in EM 420T and JEM 4000EX instruments.The (200) dark-field cross-sectional imaging is the most appropriate TEM technique to distinguish between individual layers in 111-v semiconductor heterostructures.