Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces

A new characterization of the positive self-adjoint extensions of symmetric operators, T 0 T_0 , is presented, which is based on the Friedrichs extension of T 0 , T_0, a direct sum decomposition of domain of the adjoint T 0 ∗ T_0^{*} and the boundary mapping of T 0 ∗ T_0^{*} . In applying this result to ordinary differential equations, we characterize all positive self-adjoint extensions of symmetric regular differential operators of order 2 n 2n in terms of boundary conditions.

We discuss a problem of constructing self-adjoint ordinary differential operators starting from self-adjoint differential expressions based on the general theory of self-adjoint extensions of symmetric operators outlined in [1]. We describe one of the possible ways of constructing in terms of the closure of an initial symmetric operator associated with a given differential expression and deficient spaces. Particular attention is focused on the features peculiar to differential operators, among them on the notion of natural domain and the representation of asymmetry forms generated by adjoint operators in terms of boundary forms. Main assertions are illustrated in detail by simple examples of quantum-mechanical operators like the momentum or Hamiltonian.

A positive definite eigenvalue problem for second-order self-adjoint elliptic differential operator defined on a square domain in the plane with homogeneous Dirichlet boundary condition is considered. This problem has a nondecreasing sequence of positive eigenvalues of finite multiplicity with a limit point at infinity. To the sequence of eigenvalues, there corresponds an orthonormal system of eigenfunctions. The original differential eigenvalue problem is approximated by the finite element method with numerical integration and Lagrange biquadratic finite elements. Error estimates for approximate eigenvalues and eigenfunctions are established.

An alternative linear flow model and boundary integral flow representation for ship motions in regular waves

We consider a selfadjoint elliptic operator in an infinite multidimensional cylinder with the Dirichlet boundary condition which is replaced by the Robin condition on small sets located along a given line on the boundary. The shape and distribution of these sets are arbitrary. The characteristic linear size of these sets is a small parameter of the problem. It is shown that the resolvent of such an operator converges to the resolvent of the homogenized operator, and an estimate for the convergence rate is obtained. The homogenized operator is the same operator, but without alternating boundary conditions. The difference of resolvents is estimated in the norm of bounded operators acting from L2 to $$ {W}_2^1. $$ If the small sets are periodically distributed and have the same shape and the Robin condition is replaced by the Neumann condition, we derive two-sided asymptotic estimates for the lower band functions, which provides the possibility to estimate from below the length of the first band of the spectrum.

The principal term of the spectral asymptotics is calculated for selfadjoint elliptic differential operators of second order having a strong power degeneracy on the boundary of the domain.Bibliography: 26 items.

We study Kac-Moody asymptotic symmetries and memory effects in AdS4Poincaré gauge theory and (when accompanied by 4D gravity) in its holographic CFT3 dual. While such infinite-dimensional symmetries are absent in standard asymptotic analyses of AdS4, we show how they arise with alternate AdS boundary conditions. In the 3D holographic description, these alternate boundary conditions correspond to a modified {tilde{mathrm{CFT}}}_3 obtained by Chern-Simons gauging of the CFT3 dual defined by standard boundary conditions, so that Kac-Moody symmetries then follow from the familiar Chern-Simons/Wess-Zumino-Witten correspondence. Apart from their own intrinsic interest, in abelian AdS4 gauge theories these alternate boundary conditions are equivalent to standard boundary conditions imposed on electric-magnetic dual variables. In the holographic description this corresponds to 3D “mirror” symmetries connecting the original and modified CFTs. Further, in both abelian and non-abelian theories we show that the alternative/ {tilde{mathrm{CFT}}}_3 theory emerges at leading order in large Chern-Simons level from the correlators of the standard theory, upon incorporating large-wavelength limits in the holographically emergent dimension. We point out similarities and differences between 4D AdS and Minkowski gauge theories in their asymptotic symmetries, “soft” limits and memory effects.

We consider a point interaction model for two particles in an external field, which is similar to the Skornyakov-Ter-Martirosyan model and is treated using the theory of self-adjoint extensions of symmetric operators. The corresponding energy operator has an infinite set of bound states with the energy values receding to-∞.

Application of zero-radius potentials to problems of diffraction by small inhomogeneities in elastic plates

Second-order boundary conditions are used to simulate a three-layer coating on perfectly conducting cylindrical structures. Both E and H polarizations are considered, and integral equations are presented based on the derived conditions. A moment method implementation of the resulting integral equations is given, and results are presented for various coatings in order to illustrate the accuracy of the higher-order conditions. The improvement that can be achieved with the proposed second-order generalized impedance boundary conditions over the standard boundary conditions is shown.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Numerical simulation of a turbulent bluff-body flow is conducted using large eddy simulation (LES). The open source CFD software package, OpenFOAM, is employed to solve the LES filtered transport equations governing the three-dimensional incompressible flow in the wake of the body. This work is motivated by the importance of bluff-body flows, for example, in flame stabilization in industrial combustors and burners, as well as in aerodynamics applications. The focus of the study is on the proper generation of turbulence at the inlet boundary. Standard boundary conditions available in OpenFOAM are not sufficient for providing an accurate turbulent inlet condition without modification of the bluff geometry. An improved method describing the inflow boundary condition is developed based on the existing OpenFOAM mapping-type boundary condition. In this method, the boundary condition scales the standard deviation and mean value of the velocity field onto the prescribed values provided by the experimental data. The method is implemented in OpenFOAM and employed in LES prediction of a turbulent bluff-body flow, studied in the experiments of the Clean Combustion Research Group at the University of Sydney. The LES results show favorable agreements with the experimental data.

Periodic boundary condition is commonly used in the FE homogenization method for predicting the thermo-mechanical properties of composites. However, its existing numerical algorithms are generally inefficient. Linear displacement boundary condition is much easier and more efficient to implement for the Representative Volume Elements (RVEs) with conformal and nonconformal surface meshes. Aiming at replacing the periodic boundary condition by the linear displacement boundary condition to easily and efficiently predict the thermo-mechanical properties of composites, this paper compares the implementation efficiency and prediction accuracy of the periodic and linear displacement boundary conditions. Regarding the prediction of the elastic properties of composites, the balance of micro-energy and macro-energy of the RVEs imposed the periodic and linear displacement boundary conditions is checked, and the results demonstrate that these two boundary conditions guarantee the Hill lemma (i.e. the balance of micro-energy and macro-energy of the RVEs). Moreover, comparing with the experimental test and the multi-step micro-mechanical method, the effectiveness of the periodic and linear displacement boundary conditions to predict the elastic properties of composites is validated.

This paper is devoted to the computation of certain directional semi-derivatives of eigenvalue functionals of self-adjoint elliptic operators involving a variety of boundary conditions. A uniform treatment of these problems is possible by considering them as a problem of calculating the semi-derivative of a minimum with respect to a parameter. The applicability of this approach, which can be traced back to the works of Danskin [8, 9] and Zolésio [28], to the treatment of eigenvalue problems (where the full shape derivative may not exist, due to multiplicity issues), has been illustrated by Zolésio in [29] (see also [10, Chap. 10] and included references). Despite this, some of the recent literature (see, for example, [1] or [7]) on the shape sensitivity of eigenvalue problems still continue to employ methods such as the material derivative method or Lagrangian methods which seem less adapted to this class of problems. The Delfour–Zolésio approach does not seem to be fully exploited in the existing literature: we aim to recall the importance and the simplicity of the ideas from [8, 28], by applying it to the analysis of the shape sensitivity for eigenvalue functionals for a class of elliptic operators in the scalar setting (Laplacian or diffusion in heterogeneous media), thus recovering known results in the case of Dirichlet or Neumann boundary conditions and obtaining new results in the case of Steklov or Wentzell boundary conditions.

This paper deals with fast simulations of the haemodynamics in large arteries by considering a reduced model of the associated fluid-structure interaction problem, which in turn allows an additional reduction in terms of the numerical discretisation. The resulting method is both accurate and computationally cheap. This goal is achieved by means of two levels of reduction: first, we describe the model equations with a reduced mathematical formulation which allows to write the fluid-structure interaction problem as a Navier-Stokes system with non-standard boundary conditions; second, we employ numerical reduction techniques to further and drastically lower the computational costs. The non standard boundary condition is of a generalized Robin type, with a boundary mass and boundary stiffness terms accounting for the arterial wall compliance. The numerical reduction is obtained coupling two well-known techniques: the proper orthogonal decomposition and the reduced basis method, in particular the greedy algorithm. We start by reducing the numerical dimension of the problem at hand with a proper orthogonal decomposition and we measure the system energy with specific norms; this allows to take into account the different orders of magnitude of the state variables, the velocity and the pressure. Then, we introduce a strategy based on a greedy procedure which aims at enriching the reduced discretization space with low offline computational costs. As application, we consider a realistic haemodynamics problem with a perturbation in the boundary conditions and we show the good performances of the reduction techniques presented in the paper. The results obtained with the numerical reduction algorithm are compared with the one obtained by a standard finite element method.The gains obtained in term of CPU time are of three orders of magnitude.

Bending and buckling analysis of functionally graded Timoshenko nanobeam using Two-Phase Local/Nonlocal piezoelectric integral model