Conical Domains Research Articles

Fundamental solutions and critical Lane-Emden exponents for nonlinear integral operators in cones

In this article we study the fundamental solutions or “α-harmonic functions” for some nonlinear positive homogeneous nonlocal elliptic problems in conical domains, such asF(u)=0inCω,u=0inRN∖Cω, where ω is a proper C2 domain in SN−1 for N≥2, Cω:={x:x≠0,|x|−1x∈ω} is the cone-like domain related to ω, and F is an extremal fully nonlinear integral operator. We prove the existence of two fundamental solutions that are homogeneous and do not change signs in the cone; one is bounded at the origin and the other at infinity.As an application, we use the fundamental solutions obtained to prove Liouville type theorems in cones for supersolutions of the Lane-Emden-Fowler equation in the formF(u)+up=0inCω,u=0inRN∖Cω. We also prove a generalized Hopf type lemma in domains with corners. Most of our results are new even when F is the fractional Laplacian operator.

Kinetics of the domain structure in calcium orthovanadate crystals under application of the series of electric field pulses

The evolution of as-grown domain structure under the action of the series of rectangular electric field pulses at the elevated temperature was studied by in situ optical imaging in calcium orthovanadate single crystals. It was revealed that the domain switching was accompanied by formation of the holes in the electrodes stimulated by conical domains reaching the surface. The multistage scenario of the domain kinetics was considered. It was shown that the evolution of the domain structure represented the independent growth of few large domains by motion of the smooth domain walls in the bulk and at the surface.

Seismicity near Mayotte explained by interacting magma bodies: Insights from numerical modeling

Mayotte island experienced a large volcanic eruption 50 km offshore in 2018–2021, creating the submarine volcano “Fani Maoré”. The eruption was accompanied by intense seismicity at mantle depths (20–45 km), divided into a “proximal” and a “distal” cluster centered 10 and 30 km east from the island, respectively. Previous studies suggest that two separate magma reservoirs may lie at the top and bottom of the proximal cluster. Here, we assess whether two reservoirs are a mechanically viable explanation for the proximal cluster's truncated conical shape. We developed finite-element models of pressurized magma reservoirs in a 2D axisymmetric domain, modeling the reservoirs as compliant elastic ellipsoids embedded in an elastoplastic host rock. We find that, at these depths, extremely low friction is required to generate failure at realistically low reservoir pressures. This implies in turn that mechanical weakening must occur at these depths. The weakening could be induced by fractures or pore fluid overpressure in the volcanic system. We find that two superimposed reservoirs can generate a plastic domain between them, if they are spatially close enough. Several reservoir geometries (from spherical to sill-like) are plausible. A conical fracture domain is more likely to appear for reservoirs with opposite pressure loads (i.e. one inflating, one deflating). Given the geometrical match with the proximal seismicity cluster at Mayotte, we suggest that the shallower (Moho-depth) reservoir is inflating, creating a potential hazard for Mayotte island.

Estimation of changes in the length of a moving conical domain wall in bistable microwire

The single domain wall process of magnetization reversal in a bi-stable microwire is studied. Models for static planar and folded planar domain walls are analyzed. The results obtained show that a folded domain wall can have lower total energy. Based on this finding, the dynamic properties of the more probable conical shape of a moving domain wall were studied. Using a simple model of the folded conical domain wall, the formulas for length, velocity, and mobility of the domain wall as a function of the applied magnetic field were derived. The main result of the proposed model is that it predicts domain wall shortening with an increasing applied magnetic field. Fitting the model function to the experimental data made it possible to estimate the characteristic length of the domain wall and changes in its shape in the magnetic field applied. Comparison of the model and experimental data indicates that there is a difference in eddy current damping for fast and slow domain walls, which may be responsible for the unidirectional effect in domain wall propagation.

Oblique derivative problems for elliptic equations on conical domains

We study the oblique derivative problem for uniformly elliptic equations on cone domains. Under the assumption of axi-symmetry of the solution, we find sufficient conditions on the angle of the oblique vector for Hölder regularity of the gradient to hold up to the vertex of the cone. The proof of regularity is based on the application of carefully constructed barrier methods or via perturbative arguments. In the case that such regularity does not hold, we give explicit counterexamples. We also give a counterexample to regularity in the absence of axi-symmetry. Unlike in the equivalent two-dimensional problem, the gradient Hölder regularity does not hold for all axi-symmetric solutions, but rather the qualitative regularity properties depend on both the opening angle of the cone and the angle of the oblique vector in the boundary condition.

Conical Statistical Optimal Near-Field Acoustic Holography with Combined Regularization.

For the sound field reconstruction of large conical surfaces, current statistical optimal near-field acoustic holography (SONAH) methods have relatively poor applicability and low accuracy. To overcome this problem, conical SONAH based on cylindrical SONAH is proposed in this paper. Firstly, elementary cylindrical waves are transformed into those suitable for the radiated sound field of the conical surface through cylinder-cone coordinates transformation, which forms the matrix of characteristic elementary waves in the conical spatial domain. Secondly, the sound pressure is expressed as the superposition of those characteristic elementary waves, and the superposition coefficients are solved according to the principle of superposition of wave field. Finally, the reconstructed conical pressure is expressed as a linear superposition of the holographic conical pressure. Furthermore, to overcome ill-posed problems, a regularization method combining truncated singular value decomposition (TSVD) and Tikhonov regularization is proposed. Large singular values before the truncation point of TSVD are not processed and remaining small singular values representing high-frequency noise are modified by Tikhonov regularization. Numerical and experimental case studies are carried out to validate the effectiveness of the proposed conical SONAH and the combined regularization method, which can provide reliable evidence for noise monitoring and control of mechanical systems.

Dimensionality increase of ferroelectric domain shape by pulse laser irradiation

The increase of the domain shape dimensionality from one-dimensional (1D) to two-dimensional (2D) representing creation of the magnetic bubbles by application of the uniform magnetic field to the stripe domain structure is a well-known effect used for the magnetic memory. The analogy between ferromagnetic and ferroelectric domains stimulates a search of a similar effect in ferroelectrics. Here, we present the discovered 1D to 2D transformation of the domain shape under the action of pyroelectric field appeared in lithium niobate crystal as a result of irradiation by IR laser pulses. The imaging at the surface reveals the transformation of the stripe domains appeared after the first pulse into the regular arrays of isolated circular domains after the second pulse, whereas the imaging in the bulk reveals the transformation of the comb-like domain to array of isolated conical domains. The process was considered in terms of the kinetic approach. The mechanism of the comb-like domain formation by continuous tooth generation during domain elongation is proposed. The 1D to 2D transformation is attributed to the domain splitting due to the backward motion of the charged domain wall. The proposed mechanisms are confirmed by computer simulation of the temporal dependence of the pyroelectric field excess over the threshold value.

On Some Multidimensional Limit Boundary Value Problems

We study solvability for a model elliptic pseudo-differential equation with an additional integral condition in Sobolev–Slobodetskii spaces in certain conical domains in Euclidean space. . Using the wave factorization for an elliptic symbol we construct a solution for this boundary value problem and study the behavior of the solution when parameters of cones tend to their extremal values. It was shown that for such a solvability we need certain additional condition besides presented integral condition.

Evolution of the domain structure during polarization reversal in relaxor SBN single crystals studied by Čerenkov-type second harmonic generation microscopy

The evolution of the domain structure during polarization reversal in single crystals of a relaxor ferroelectric strontium-barium niobate (Sr0.61Ba0.39Nb2O6) doped with 0.05 wt.% Ni2O3 was investigated. The domains in the crystal bulk were imaged by Čerenkov-type second harmonic generation microscopy. Three types of domains were revealed: (1) domains with a broad boundary, (2) cylindrical domains, and (3) conical domains. Three stages of the domain shape evolution were distinguished in the domain with a broad boundary. The velocities of the sideways wall motion and forward domain growth were measured for all the domain types.

Self-formation of dual glassy-crystalline structure in magnetron sputtered W–Zr films

Self-formation of a unique dual glassy-crystalline structure in binary W–Zr system was observed for a film with 28 at.% Zr prepared by magnetron co-sputtering. The film is composed of conical columnar domains of α-W(Zr) solid solution structure surrounded by featureless areas corresponding to a W–Zr metallic glass. The conical domains have their axes perpendicular to the film surface. Most of the domains have the apex at or close to the substrate surface, which corresponds to the point of primary nucleation. The surface ratio of glassy and crystalline phase (bases of the cones) is dependent on the film thickness. The dual structure is prepared in a very narrow window of the elemental composition. We suppose that the specific elemental composition and the diffusivity or mobility of sputtered adatoms are crucial for the self-formation of the dual structure.

Self-Formation of Dual Glassy-Crystalline Structure in Magnetron Sputtered W–Zr Films

Self-formation of dual glassy-crystalline structure in binary W–Zr system was observed for a film with 28 at.% Zr prepared by magnetron co-sputtering. The film is composed of conical columnar domains of α-W(Zr) solid solution structure surrounded by featureless areas corresponding to a W–Zr metallic glass. The conical domains have their axes perpendicular to the film surface. Most of the domains have the apex at or close to the substrate surface, which corresponds to the point of primary nucleation. The surface ratio of glassy and crystalline phase (bases of the cones) is dependent on the film thickness. The dual structure is prepared in a very narrow window of the elemental composition. We suppose that the specific elemental composition and the diffusivity or mobility of sputtered adatoms are crucial for the self-formation of the dual structure.

Krasnosel’skii type compression-expansion fixed point theorem for set contractions and star convex sets

In this paper, we give or improve compression-expansion results for set contractions in conical domains determined by balls or star convex sets. In the compression case, we use Potter’s idea of proof, while the expansion case is reduced to the compression one by means of a change of variable. Finally, to illustrate the theory, we give an application to the initial value problem for a system of implicit first order differential equations.

Subclasses of analytic functions with respect to symmetric and conjugate points bounded by conical domain

The main objective of this present investigation is to derive the Fekete Szeg$\ddot{o}$ inequality for the function belongs to the defined class in the open unit disk with respect to symmetric and conjugate points bounded by conical region. As a special consequences of our result, we extracted some interesting results through corollaries.

On Certain 3-Dimensional Limit Boundary Value Problems

The paper is devoted to studying limit behavior for a solution of model elliptic pseudo-differential equation with some integral boundary condition in 4-wedge conical canonical 3D singular domain with two parameters. It is shown that the solution of such boundary value problem can have a limit with respect to endpoint values of the parameters in appropriate Sobolev–Slobodetskii space if the boundary function is a solution of a special functional singular integral equation.

Study of 2m-th order parabolic equation in non-symmetric conical domains

This article is devoted to the study of a $N$-space dimensional linear high-order parabolic equation, subject to Cauchy-Dirichlet boundary conditions. The problem is set in a non-symmetric conical domain. The analysis is performed in the framework of weighted anisotropic Sobolev spaces by using the domain decomposition method.

Hypersurfaces with free boundary in a convex conical domain in a space form

Hypersurfaces with free boundary in a convex conical domain in a space form

Bounds of the initial coefficient for sakaguchi function in the conical domain

In this paper, we consider a new class of sakaguchi type functions which is defined by Ruscheweyh q-differential operator. We investigated of co-efficient inequalities and other interesting properties of this class.

The Helmholtz decomposition in weighted $L^p$ spaces in cones

In this paper we prove the Helmholtz decomposition in conical domains of $\mathbb{R}^3$ in weighted $L^p_\beta$ spaces under a spectral condition on $\beta,p$. The basic elements are the transformation of the original problem into a problem set in cylindrical domains and the combination of a priori bounds from [5] with the vector-valued multiplier theorem [29].

A representative linear eddy model for simulating spray combustion in engines (RILEM)

The design of new combustion concepts for low emission, high efficiency internal combustion engines often leads to combustion under low temperature conditions. Under those conditions, the assumption of fast chemistry, which has been the cornerstone of many turbulent combustion models, is not strictly valid anymore and the validity and applicability of classical combustion models such as flamelet models might be limited. In this paper we present an updated version of a recently developed regime independent modeling approach for turbulent non-premixed combustion with an emphasis on applications to internal combustion engines. The model utilizes the mode- and regime-independent linear eddy model (LEM) as a combustion and micro-mixing model in a representative way. This is achieved by time advancing only one LEM realization representing the combustion process in the whole engine domain and coupling it to a RANS simulation with a presumed β-function PDF approach for the mixture fraction. The use of LEM rather than flamelet combustion closure has several benefits, an important one being regime independence. Additionally, LEM incorporates a physically based representation of the stochastic variability of turbulent eddy motions, implying an intrinsic representation of scalar dissipation rate fluctuations. In order to capture key features of engine spray-combustion environments, the LEM methodology is extended by introducing a conical LEM domain to approximate spray spatial development, fuel vapor input based on CFD-prescribed spray evaporation, and a representation of large scale turbulent motions distinct from the inertial-range turbulence that develops at smaller scales. The representative character of LEM states is evaluated by comparing mixture fraction statistics and scalar dissipation rates generated by LEM and the CFD. The performance and predictive capability of the model for typical engine applications is evaluated by simulating a standard test case – Spray B of the Engine Combustion Network (ECN) – and comparing the results with experimental data. The results demonstrate the capability of the model to represent the spray combustion process with reasonable accuracy but also reveal some limitations. The limitations and shortcomings of the model are discussed and an outlook for further development of the approach into a regime- and mode-independent combustion model for internal engine applications is given.

Starlike and convex functions with respect to symmetric conjugate points involving conical domain

AbstractEnough attentions to domains related to conical sections has not been done so far although it deserves more. Making use of the conical domain the authors have defined a new class of starlike and Convex Functions with respect to symmetric points involving the conical domain. Growth and distortion estimates are studied with convolution using domains bounded by conic regions. Certain coefficient estimates are obtained for domains bounded by conical region. Finally interesting application of the results are also highlighted for the function Ωk,βdefined by Noor.