The Finite Spectrum of Sturm–Liouville Problems with Transmission Conditions and Eigenparameter-Dependent Boundary Conditions

We study the finite spectrum of Sturm–Liouville problems with transmission conditions and eigenparameter-dependent boundary conditions. For any positive integers m and n, we construct a class of regular Sturm–Liouville problems with transmission conditions and eigenparameter-dependent boundary conditions, which have at most m + n + 4 eigenvalues.     

Stationary Solutions of Compressible Navier–Stokes Equations with Slip Boundary Conditions

We consider the Navier–Stokes equations for a compressible, viscous fluid with heat–conduction in a bounded domain of IR² or IR³. Under the assumption that the external force field and the external heat supply are small we prove the existence and local uniqueness of a stationary solution satisfying a slip boundary condition. For the temperature we assume a Dirichlet or an oblique boundary condition.     

Theorem of Completeness for a Dirac-Type Operator with Generalized λ-Depending Boundary Conditions

A completeness theorem is proved involving a system of integro-differential equations with some λ-depending boundary conditions. Also some sufficient conditions for the root functions to form a Riesz basis are established. The research was supported by the Academy of Finland (project 129092).     

Convergence Analysis of the Nonconforming Finite Element Discretization of Stokes and Navier–Stokes Equations with Nonlinear Slip Boundary Conditions

This work is concerned with the nonconforming finite approximations for the Stokes and Navier–Stokes equations driven by slip boundary condition of “friction” type. It is well documented that if the velocity is approximated by the Crouzeix–Raviart element of order one, whereas the discrete pressure is constant elementwise that the inequality of Korn does not hold. Hence, we propose a new formulation taking into account the curvature and the contribution of tangential velocity at the boundary. Using the maximal regularity of the weak solution, we derive a priori error estimates for the velocity and pressure by taking advantage of the enrichment mapping and the application of Babuska–Brezzi’s theory for mixed problems.     

A Differential Boundary Operator of the Sturm–Liouville Type on a Semiaxis with Two-Point Integral Boundary Conditions

We prove that a differential boundary operator of the Sturm–Liouville type on a semiaxis with two-point integral boundary conditions that acts in the Hilbert space L ₂(0, ) is closed and densely defined. The adjoint operator is constructed. We also establish criteria for the maximal dissipativity and maximal accretivity of this operator.     

The Existence of Solutions to Sturm-Liouville Boundary Value Problems with Laplacian-like Operator

Abstract The existence of infinitely many solutions to Sturm-Liouville boundary value problem with aLaplacian-like operator is studied by applying generalized polar coordinates.     

Solvability of the Navier—Stokes System with L 2 Boundary Data

We prove the existence of the very weak solution of the Dirichlet problem for the Navier—Stokes system with L ² boundary data. Under the small data assumption we also prove the uniqueness. We use the penalization method to study the linearized problem and then apply Banach's fixed point theorem for the nonlinear problem with small boundary data. We extend our result to the case with no small data assumption by splitting the data on a large regular and small irregular part. Accepted 15 March 1999     

Direct and Inverse Problems for the Matrix Sturm–Liouville Operator with General Self-Adjoint Boundary Conditions

Mathematical Notes - The matrix Sturm–Liouville operator on a finite interval with boundary conditions in general self-adjoint form and with singular potential of class $$W_2^{-1}$$ is...     

Spectrum and Spectral Singularities of a Quadratic Pencil of a Schr?dinger Operator with Boundary Conditions Dependent on the Eigenparameter

In this paper, we consider the following quadratic pencil of Schr?dinger operators L(λ)generated in L²(R⁺) by the equation ■ with the boundary condition ■ where p(x) and q(x) are complex valued functions and α₀, α₁, β₀, β₁ are complex numbers with α₀β₁-α₁β₀≠0. It is proved that L(λ) has a finite number of eigenvalues and spectral singularities,and each of them is of a finite multiplicity...     

Basis Properties of the System of Root Functions of the Sturm–Liouville Operator with Degenerate Boundary Conditions: I

The spectral problem for the Sturm–Liouville operator with an arbitrary complex-valued potential q(x) of the class L¹(0, π) and degenerate boundary conditions is considered. We prove that the system of root functions of this operator is not a basis in the space L²(0, π).     

Stationary solutions of the Navier-Stokes equations in the spaces L p (ℝ n )

We study the existence and regularity of solutions of the stationary Navier-Stokes system in the spaces L _p (? ⁿ ). The use of the theory of multipliers of the Fourier transform permits one to single out a class of spaces in which there exists a unique “small” solution. We study the regularity of solutions in these spaces without the smallness assumption.     

Heat Kernel Estimates for Schrödinger Operators on Exterior Domains with Robin Boundary Conditions

We study Schrödinger operators with Robin boundary conditions on exterior domains in ? ^d . We prove sharp point-wise estimates for the associated semigroups which show, in particular, how the boundary conditions affect the time decay of the heat kernel in dimensions one and two. Applications to spectral estimates are discussed as well.     

The Conditions for the Existance of n Limit-Cycles in Lienard Equation

The Lienard equation is For this equation we assume 1) The function f(x) and g(x) are continuous on where 2) There is a sequance sa tisying that And we have where i = 0, 1,2,…, n-2; and (Take the similar gn in the following conditions).     

Characterization and Connectivity of Generalized Filters in L 2(ℝ d )

Let A be a d×d real expansive integer matrix (i.e., a matrix with real entries whose eigenvalues are all of modules greater than one) with |det?A|=2, and let m (which is called A-dilation generalized filter) be a 2π? ^d periodic function with the property that |m(s)|²+|m(s+2π h ₂)|²=1, where h ₂∈(A ^τ )^?1? ^d ?? ^d . In this paper, we characterize the set of all A-dilation generalized filters and show that this set is path-connected in $L^{2}({\mathbb{T}}^{d})$ -norm by using the technique of filter multipliers. We also obtain an equivalent condition for an A-dilation generalized filter to be an A-dilation low pass filter. These extend the results of Manos Papadakis et al. from one dimensional case to high dimensions and matrix dilations cases.     

Resolvent Operator and Self-Adjointness of Sturm–Liouville Operators with a Finite Number of Transmission Conditions

In this paper, we consider a Sturm–Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions at a finite number of interior points. We introduce a Hilbert space formulation such that the problem under consideration can be interpreted as an eigenvalue problem for a suitable self-adjoint linear operator. We construct Green function of the problem and resolvent operator. We establish the self-adjointness of the discontinuous Sturm–Liouville operator.     

Average Widths of Sobolev-Wiener Classes with Mixed Smoothness in L q (ℝ d )

In this paper, the average σ-Κ width of Sobolev-Wiener classes S ^r _pq W with mixed smoothness in L _q (ℝ ^d ) is studied for 1 < q≤p < ∞, and the weak asymptotical behaviour of these quantities is obtained. Received July 21, 1998, Accepted May 31, 1999     

The Riesz Basis Property of an Indefinite Sturm–Liouville Problem with Non-Separated Boundary Conditions

We consider a regular indefinite Sturm–Liouville eigenvalue problem ?f′′ + q f = λ r f on [a, b] subject to general self-adjoint boundary conditions and with a weight function r which changes its sign at finitely many, so-called turning points. We give sufficient and in some cases necessary and sufficient conditions for the Riesz basis property of this eigenvalue problem. In the case of separated boundary conditions we extend the class of weight functions r for which the Riesz basis property can be completely characterized in terms of the local behavior of r in a neighborhood of the turning points. We identify a class of non-separated boundary conditions for which, in addition to the local behavior of r in a neighborhood of the turning points, local conditions on r near the boundary are needed for the Riesz basis property. As an application, it is shown that the Riesz basis property for the periodic boundary conditions is closely related to a regular HELP-type inequality without boundary conditions.