Annular Domain Research Articles

RADIAL SOLUTION OF ASYMPTOTICALLY LINEAR ELLIPTIC EQUATION WITH MIXED BOUNDARY VALUE IN ANNULAR DOMAIN

In this paper, we study nonlinear elliptic equation with mixed boundary value condition in annular domain. It is assumed that the nonlinearity is asymptotically linear and depends on the derivative term. Some results on the existence of solution are established by nonlinear analysis methods.

Radial solutions of Lane-Emden-Fowler equations with Pucci's extremal operators

We report on some recent results obtained for positive radial solutions of Lane-Emden-Fowler type equations with Pucci's operators as principal parts. The presented results include the asymptotic analysis of almost critical solutions in the unit ball, existence results in annular domains and sharp Liouville-type results for exterior Dirichlet problems.

Symmetry results for viscosity solutions of fully nonlinear equations in annular and exterior domains

In this paper, we prove the symmetry and monotonicity results of viscosity solutions for fully nonlinear elliptic equations F(D2u,Du,u,x)=0 and fully nonlinear parabolic equations −ut+F(D2u,Du,u,x,t)=0 in annular domains by using the method of moving planes. We also prove the symmetry and monotonicity results of viscosity solutions for fully nonlinear parabolic equations −ut+F(D2u,Du,u,t)=0 in exterior domains. This extends the results of fully nonlinear elliptic equations in exterior domains.

Effect of the horizontal aspect ratio on thermocapillary convection stability in annular pool with surface heat dissipation

A linear stability analysis of the thermoconvective problem of a thin liquid film contained in an annular domain has been conducted. The influence of the horizontal aspect ratio on the solution has been considered by keeping a fixed external wall while the internal radius of the annular domain was modified. The parameter used in the study, Γh, has been defined as the ratio of the internal radius to the domain depth. The other control parameter of the study is the Prandtl number ranging from 0.7 to 50, i.e. characteristic of fluids as air to n−butanol. The study has been performed for different Bond (Bo) regimes ranging from 0.0 for surface tension dominated flows to 67 for buoyancy dominated ones. Three different kind of bifurcations are found in the Γh−Pr plane for large Bonds, while for low Bonds only two of them appear. In the case of pure buoyancy or surface tension flows, for every Γh there exists a Prandtl number such that oscillatory and stationary coexist in a co-dimension two bifurcation point. These transitions show a strong dependency with the Bond number. Indeed, the lower transition disappears for low Bo and the upper one disappears with intermediate Bo values. Furthermore, there is a non-linear dependency of the number of structures of the growing bifurcation with Γh. These co-dimension two lines show a strong dependency with Bo. Firstly, looking at the frontier between HWI and LR regions, for large Bo numbers, Pr increases with Γh, while for low Bo the trend is reversed. Additionally, this transition only appears in the extreme Bo cases, for the central values of the considered, no transition is found. Similarly, the second transition found only appears for Bo larger than 30.

Entanglement entropy of an annulus in holographic thermalization * * Supported by the Natural Science Foundation of China (11575195, 11875053 (Y.L.), 11847229 (Z.-Y.X.)), Y.L. also acknowledges the support from 555 talent project of Jiangxi Province. Z.-Y.X. also acknowledges the support from the National Postdoctoral Program for Innovative Talents (BX20180318)

The thermalization process of the holographic entanglement entropy (HEE) of an annular domain is investigated in the Vaidya-AdS geometry. We determine numerically the Hubeny-Rangamani-Takayanagi (HRT) surface, which may be a hemi-torus or two disks, depending on the ratio of the inner radius to the outer radius of the annulus. More importantly, for some fixed ratio of the two radii, the annulus undergoes a phase transition, or a double phase transition, during thermalization from a hemi-torus to a two-disk configuration, or vice versa. The occurrence of various phase transitions is determined by the ratio of the two radii of the annulus. The rate of entanglement growth is also investigated during the thermal quench. The local maximal rate of entanglement growth occurs in the region with a double phase transition. Finally, if the quench process is sufficiently slow, which may be controlled by the thickness of the null shell, the region with a double phase transition vanishes.

Unique determination of inverse electromagnetic scattering by a two-layered cavity

This paper is concerned with the inverse electromagnetic scattering by a two-layered medium in the non-magnetic case. We prove that both the electric permittivity and the conductivity can be uniquely recovered in an unknown annular domain by the internal wave-field measurements at one fixed frequency, if and are both assumed to be constant. Furthermore, we prove that the unknown annular domain can be also uniquely recovered using the same measurements. The method proposed in this paper essentially depends on the uniform a priori estimate for the scattered magnetic field in the sense when the incident fields are induced by a sequence of deformed electric dipoles. Another important ingredient for our method is to reconstruct well-defined PDE systems in a sufficiently small domain by using the solutions to the electromagnetic scattering problem in the two-layered cavity.

On overdetermined problems for a general class of nonlocal operators

We study the overdetermined problem for a large family of non-local operators given by generators of subordinate Brownian motions. In particular, this family includes the fractional Laplacian, relativistic stable operators etc. We consider these problems in bounded domains, exterior domains, and in annular domains and we show that under suitable conditions, the domains and solutions are both radially symmetric. Our method uses both analytic and probabilistic tools.

On a boundary integral solution of a lateral planar Cauchy problem in elastodynamics

A boundary integral based method for the stable reconstruction of missing boundary data is presented for the governing hyperbolic equation of elastodynamics in annular planar domains. Cauchy data in the form of the solution and traction is reconstructed on the inner boundary curve from the similar data given on the outer boundary. The ill-posed data reconstruction problem is reformulated as a sequence of boundary integral equations using the Laguerre transform with respect to time and employing a single-layer approach for the stationary problem. Singularities of the involved kernels in the integrals are analyzed and made explicit, and standard quadrature rules are used for discretization. Tikhonov regularization is employed for the stable solution of the obtained linear system. Numerical results are included showing that the outlined approach can be turned into a practical working method for finding the missing data.

Wave propagation from lateral Cauchy data using a boundary element method

The wave equation with lateral Cauchy data is considered in three-dimensional annular domains. It is a linear inverse ill-posed problem, where data in the form of the function values and normal derivative is given on the outer boundary surface. To reconstruct the missing data on the inner surface a recent boundary integral approach is applied and extended to three-dimensions. A transformation in time (semi-discretization) is employed via the Laguerre transform resulting in a Cauchy problem for a system consisting of a sequence of inhomogeneous elliptic equations. The solution to this system is represented as a sequence of single-layer potentials invoking a fundamental sequence of the elliptic equations, with the benefit of avoiding the use of any volume potentials or domain discretization. Matching the data against the single-layer representation of the solution gives a system of boundary integral equations to be solved for a sequence of densities. A Galerkin boundary element discretization is applied to this latter system, triangulating the boundary surfaces and using piecewise constant approximating functions. Tikhonov regularization is used when solving the sequence of discrete linear equations. Numerical examples show the feasibility of the proposed approach rendering accurate and stable approximations of the missing lateral data with little computational effort.

Critical points of solutions for the mean curvature equation in strictly convex and nonconvex domains

In this paper, we mainly investigate the set of critical points associated to solutions of mean curvature equation with zero Dirichlet boundary condition in a strictly convex domain and a nonconvex domain respectively. Firstly, we deduce that mean curvature equation has exactly one nondegenerate critical point in a smooth, bounded and strictly convex domain of $\mathbb{R}^{n}(n\geq2)$. Secondly, we study the geometric structure about the critical set $K$ of solutions $u$ for the constant mean curvature equation in a concentric (respectively an eccentric) spherical annulus domain of $\mathbb{R}^{n}(n\geq3)$, and deduce that $K$ exists (respectively does not exist) a rotationally symmetric critical closed surface $S$. In fact, in an eccentric spherical annulus domain, $K$ is made up of finitely many isolated critical points ($p_1,p_2,\cdots,p_l$) on an axis and finitely many rotationally symmetric critical Jordan curves ($C_1,C_2,\cdots,C_k$) with respect to an axis.

Two-dimensional simulation of the damped Kuramoto–Sivashinsky equation via radial basis function-generated finite difference scheme combined with an exponential time discretization

We apply a numerical scheme based on a meshless method in space and an explicit exponential Runge-Kutta in time for the solution of the damped Kuramoto–Sivashinsky equation in two-dimensional spaces. The proposed meshless method is radial basis function-generated finite difference, which approximates the derivatives of the unknown function with respect to the spatial variables by a linear combination of the function values at given points in the domain and weights. Also, in this approach there is no need a mesh or triangulation for approximation. For each point, the weights are computed separately in its local sub-domain by solving a small radial basis function interpolant. Besides, a numerical algorithm based on singular value decomposition of the local radial basis function interpolation matrix [59] is applied to find the suitable shape parameter for each interpolation problem. We also consider an explicit time discretization based on exponential Runge–Kutta scheme such that its stability region is bigger than the classical form of Runge-Kutta method. Some numerical simulations are provided on the square, circular and annular domains to show the capability of the numerical scheme proposed here.

Positive Solutions to Nonlinear Elliptic Equations Depending on a Parameter with Dirichlet Boundary Conditions

The authors prove some new results on the existence of positive radial solutions to the elliptic equation $$-\Delta u= \lambda h(|x|,u)$$ in an annular domain in $${\mathbb {R}}^{N}, N\ge 2$$ . Existence of positive radial solutions are determined under the conditions that the nonlinearity h(t, u) is of either superlinear or sublinear growth in u or satisfies some upper and lower inequalities on h. Their approach is based on a revised version of a fixed point theorem of Gustafson and Schmitt.

Weakly nonlinear mode interactions in spherical Rayleigh–Bénard convection

In an annular spherical domain with separation $d$, the onset of convective motion occurs at a critical Rayleigh number $Ra=Ra_{c}$. Solving the axisymmetric linear stability problem shows that degenerate points $(d=d_{c},Ra_{c})$ exist where two modes simultaneously become unstable. Considering the weakly nonlinear evolution of these two modes, it is found that spatial resonances play a crucial role in determining the preferred convection pattern for neighbouring modes $(\ell ,\ell \pm 1)$ and non-neighbouring even modes $(\ell ,\ell \pm 2)$. Deriving coupled amplitude equations relevant to all degeneracies, we outline the possible solutions and the influence of changes in $d,Ra$ and Prandtl number $Pr$. Using direct numerical simulation (DNS) to verify all results, time periodic solutions are also outlined for small $Pr$. The $2:1$ periodic signature observed to be general for oscillations in a spherical annulus is explained using the structure of the equations. The relevance of all solutions presented is determined by computing their stability with respect to non-axisymmetric perturbations at large $Pr$.

Sign changing bubbling solutions for a critical Neumann problem in 3D

Let Ω be a smooth bounded domain in R3. We consider the problem Δu−λu+|u|4u=0inΩ,with zero Neumann boundary conditions, and investigate the existence of sign changing solutions, which concentrate simultaneously around k different points of Ω as λ approaches a special value λ0. We characterize the number λ0 in terms of the Green function of −Δ+λ. In particular, we construct a family of solutions exhibiting bubbling behavior at two different points of an annular domain as λ tends to λ0. In doing so, we also derive a representation formula of the Green function of −Δ with Neumann boundary conditions in the annulus in terms of zonal harmonics.

Landing gear interwheel tonal noise characterization with the Boundary Element Method

Landing gear noise is mainly broadband in nature, but it can also feature some tonal contributions. Tonal noise significantly increases noise annoyance and can originate from facing cavities in landing gear wheels. The present work aims at using the Boundary Element Method (BEM) with a simple harmonic point source model to characterize the resonance between the two facing cylindrical cavities in the wheels of the generic nose landing gear LAGOON, where a flow independence of the tonal noise emission has been reported experimentally. Three configurations of increasing complexity are successively considered: a single wheel, the two facing wheels linked by an axle, and finally the whole landing gear, including the main strut. In terms of resonant frequencies, the BEM is shown to give results in very good agreement with analytical formulae and CFD/CAA computations from the literature. Some acoustic measurements are also performed for the whole LAGOON configuration and a very good agreement is observed between BEM results and the experimental database. The directivity of the resonant modes is detailed for each case and compared with success to CFD/CAA results, except for convective amplification effects. The amplification of each resonant mode is quantified through a so-called scattering factor and it is shown that the facing cavities present much sharper resonances than the single cavity, and that the presence of the main strut only increases the amplification of the axisymmetric mode. Finally, the modelisation of the interwheel space by an annular duct is discussed and a parametric study is conducted on the axle diameter. The results of this parametric study shed light on the annular duct model domain of validity while demonstrating the ability of the BEM to treat geometries of arbitrary complexity with low computational cost when analytical models are unavailable.

A new approach for electro-elastic analysis of piezoelectric fiber composites with arbitrary shaped inclusions under anti-plane shear and in-plane electric loadings

Electro-elastic analysis of piezoelectric fiber composites with arbitrary shaped inclusions subjected to far field uniform anti-plane shear and in-plane electric loadings is performed in this study in order to increase the electromechanical performance and protect the vulnerable piezoelectric inclusion from electromechanical damage. A three-phase model with a representative volume element of piezoelectric fiber composites with arbitrary shaped inclusions is proposed based on the generalized self-consistent theory. Meanwhile, a new conformal mapping method based on Cauchy integral and series expansion is also proposed to map an arbitrary two-connected domain into an annular domain. Both of the electro-elastic responses and the effective moduli of the piezoelectric fiber composites with arbitrary shaped inclusions for the anti-plane problem are obtained using the complex potentials in series form and the homogenization strategy. Comparisons with literature show a good agreement. Additionally, the effects of the fiber shape and the fiber volume fraction on the normal shear stress and the tangential electric field distributions outside the piezoelectric fiber, and the effective moduli are investigated using the proposed model and method. It is shown that the normal shear stresses and the tangential electric fields outside piezoelectric fiber have intense fluctuations at the corner of the inclusion. Both of the electro-elastic responses and the effective moduli of piezoelectric fiber composites become larger as the shape of piezoelectric inclusion sharpens and the fiber volume fraction increases. Furthermore, the proposed model and method can be used for determining the appropriate shape and volume fraction of the piezoelectric inclusion to significantly reduce stress and electric field concentrations around the inclusion and optimize the electromechanical performance of piezoelectric fiber composites.

On the alternating method and boundary-domain integrals for elliptic Cauchy problems

A numerical implementation of the alternating iterative method is presented for the Cauchy problem of the conductivity equation in planar annular domains. The conductivity is space dependent as opposed to earlier works, where it is assumed constant. In the constant conductivity case, the mixed problems in the iterations are solved efficiently using boundary integrals. Following on to this, for a space dependent conductivity, it is outlined how to solve these mixed problems using a recent boundary-domain integral equations approach involving the parametrix. For a mixed problem the solution is written as a combination of boundary integrals and a domain integral, with densities to be determined. The densities needed over the boundary and domain are identified by matching against the given mixed boundary data and by requiring the governing equation to hold in the domain. An efficient Nyström scheme is applied for the discretisation. Numerical results are presented for several domains and conductivities, using exact as well as noisy Cauchy data, showing that a stable solution can be obtained with good accuracy and small computational cost.

The radial basis function-differential quadrature method for elliptic problems in annular domains

Abstract We employ a radial basis function (RBF) - differential quadrature (DQ) method for the numerical solution of elliptic boundary value problems in annular domains. With an appropriate selection of collocation points, for any choice of RBF, both the coefficient and right hand side matrices in the systems appearing in this discretization possess block circulant structures. These linear systems can thus be solved efficiently using matrix decomposition algorithms (MDAs) and fast Fourier transforms (FFTs). In particular, we consider problems governed by the Poisson equation, the inhomogeneous biharmonic equation and the inhomogeneous Cauchy–Navier equations of elasticity. In addition to its simplicity, the proposed method can both achieve high accuracy and solve large-scale problems. The feasibility of the proposed techniques is illustrated by several numerical examples.

Approximate Robust Output Regulation of Boundary Control Systems

We extend the internal model principle for systems with boundary control and boundary observation, and construct a robust controller for this class of systems. However, as a consequence of the internal model principle, any robust controller for a plant with infinite-dimensional output space necessarily has infinite-dimensional state space. We proceed to formulate the approximate robust output regulation problem and present a finite-dimensional controller structure to solve it. Our main motivating example is a wave equation on a bounded multidimensional spatial domain with force control and velocity observation at the boundary. In order to illustrate the theoretical results, we construct an approximate robust controller for the wave equation on an annular domain and demonstrate its performance with numerical simulations.

Global solvability and regularity for ∂¯ on an annulus between two weakly convex domains which satisfy property (P)

Let [Formula: see text] be a complex manifold of dimension [Formula: see text] and let [Formula: see text]. Let [Formula: see text] be a weakly [Formula: see text]-convex and [Formula: see text] be a weakly [Formula: see text]-convex in [Formula: see text] with smooth boundaries such that [Formula: see text]. Assume that [Formula: see text] and [Formula: see text] satisfy property [Formula: see text]. Then the compactness estimate for [Formula: see text]-forms [Formula: see text] holds for the [Formula: see text]-Neumann problem on the annulus domain [Formula: see text]. Furthermore, if [Formula: see text] is [Formula: see text]-closed [Formula: see text]-form, which is [Formula: see text] on [Formula: see text] and which is cohomologous to zero on [Formula: see text], the canonical solution [Formula: see text] of the equation [Formula: see text] is smooth on [Formula: see text].