bluuebunny/crossref_metadata_2025_split huggingface dataset:
Generalized Hopf bifurcation in Hilbert space
Authors: Kielhöfer H. , Kirchgässner K. 
Published: March 2007

AbstractWe consider a family of semilinear evolution equations in Hilbert space of the form

equation image

with, in general, unbounded operators *A(λ), F(λ·) depending analytically on a real parameter λ. We assume that the origin is a stationary solution, i.e. F(λ,0) = 0, for all λ ε R and that the linearization (with respect to u) at the origin is given by du/dt + A(λ)u = 0. Our essential assumption is the following:
A(λ) possesses one pair of simple complex conjugate eigenvalues μ(λ) = Re μ(λ) ± i Im μ(λ) such that Im μ(0) > 0 and

equation image

for some m ε N or

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If m = 1 the curves of eigenvalues μ(λ) cross the imaginary axis transversally at ±i Im μ(0). In this case a unique branch of periodic solutions emanates from the origin at λ = 0 which is commonly called Hopf bifurcation. If μ(λ) and the imaginary axis are no longer transversal, i.e. m > 1, we call a bifurcation of periodic solutions, if it occurs, a generalized Hopf bifurcation. It is remarkable that up to m such branches may exist. Our approach gives the number of bifurcating solutions, their direction of bifurcation, and its asymptotic expansion. We regain the results of D. Flockerzi who established them in a completely different way for ordinary differential equations.
Constructive existence theorems for problems of Thomas‐Fermi Type
Authors: Mooney J. W. , Roach G. F. 
Published: March 2007

AbstractA minimal positive solution of the Thomas‐Fermi problem ẅ = λt−1/2 w3/2, w(0) = 1, w(1) = w(1) is shown to exist for each λ > 0. It is proved that all positive solutions, for a given value of λ, are strictly ordered and that the minimal positive solution wλ is a decreasing function of λ. Upper and lower analytic bounds for wλ are given and these bounds are shown to initiate sequences of Picard and Newton iterates which converge monotonically to wλ. A comparative analysis of the efficiency of the iteration schemes is presented. The methods used are of a general nature and can be applied to a variety of nonlinear boundary value problems of convex type [14].
On minimal nonlinearities which permit bifurcation from the continuous spectrum
Authors: Küpper T. , Roach G. F. 
Published: March 2007

AbstractBifurcation from the continuous spectrum of a linearized operator is of interest in many physical problems. For example it occurs in the nonlinear Klein‐Gordon equation and in nonlinear integrodifferential equations as the Choquard problem; it further appears in nonlinear integral equations of the convolution type.A general theory enclosing all these problems is not yet known. To understand the basic phenomena, we therefore consider monotone differential operators whose linearisations have a purely continuous spectrum. It is shown that in fact the lowest point of the continuous spectrum is a bifurcation point, if the nonlinearity grows sufficiently strong.
A finite element method for the solution of a potential theory integral equation
Authors: Friedman M. J. , Nedelec J. C. 
Published: March 2007

AbstractThis paper discusses a finite element approximation for an integral equation of the second kind deduced from a potential theory boundary value problem in two variables. The equation is shown to admit a unique solution, to be variational and coercive in the Hilbert space of functions σ ε H1/2(Γ), frd γ = 0. The Galerkin method with finite elements as trial functions is shown to lead to an optimal rate of convergence.
Zwei Klassen vollständiger Funktionensysteme zur Behandlung der Randwertaufgaben der Schwingungsgleichung ΔU + k2U = 0
Authors: Müller C. , Kersten H. , Leis R. 
Published: March 2007

AbstractIn this paper we present new methods to solve the classical Dirichlet and Neumann problems for ΔU + k2U = 0. We prove that the solutions of this equation for a region S containing G restricted to G are dense in L2(∂G). Introducing a basis in the space of solutions for S we find a complete orthogonal system in L2(∂G) which can be used to solve the boundary value problems by means of approximation in the Hilbertspace norm. Regularity estimates lead to series expansions in G.The well‐known basis systems obtained by separation of variables thus may be used for every regular region without the very special geometric restrictions. Another class of basis systems may be obtained in analogy to the Runge. theorems by considering types of singularity functions.
A singular perturbation approach for a reaction diffusion system arising in nuclear reactor theory
Authors: de Mottoni P. , Tesei A. , Eckhaus W. 
Published: March 2007

AbstractThe qualitative behavior of the solutions of a reaction‐diffusion system arising in the theory of nuclear reactors is investigated by means of singular perturbation techniques, the small parameter ϵ representing the inverse of neutron velocity. At the lowest approximation for small ϵ the solutions exhibit oscillations about the unique strictly positive equilibrium, much as in the case of the associated lumped parameter system. A two‐times representation for solutions of small amplitudes is also given.
Uniform convergence of the horizontal line method for solutions and free boundaries in Stefan evolution inequalities
Authors: Jerome J. W. , Brosowski B. 
Published: March 2007

AbstractThe change of variable for the temperature Θ in the one‐phase Stefan problem leads to the evolution inequality, (ut – Δu – f)(v – u) ⩾ 0 for all regular v ⩾ 0, where u ⩾ 0 is required. This inequality is to hold over a space‐time domain D = Ω × (0, T) with a Dirichlet boundary condition imposed on ∂ Ω × (0, T) and a zero initial condition. The free boundary phase interface is given in one space dimension by

The fully implicit divided difference scheme

leads to a sequence of elliptic variational inequalities for {um}. The sequence {um} may be interpolated linearly in t to obtain an approximation UΔt of u. The following results are obtained in this paper: (i) a two‐sided weak maximum principle for um – um‐1 in N space dimensions, hence the free boundary approximation for N = 1,

is a monotone increasing step function; (ii) the uniform convergence of UΔt and ∇UΔt, to u and ∇u, respectively, on D; (iii) the uniform convergence to the Hölder continuous, monotone increasing free boundary x on [0, T] of the piecewise linear sequence xΔt, where xΔt interpolates xΔt, in one space dimension; (iv) a constructive existence proof for u and x in prescribed regularity classes.
On uniqueness of Neumann‐Tricomi problem in R2
Authors: Aziz A. K. , Schneider M. , Gilbert R. P. 
Published: March 2007

AbstractWe consider the equation of mixed type

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(k(y) ⪌ 0 whenever y ⪌ 0) in a region G which is bounded by the curves: A piecewise smooth curve Γ lying in the half‐plane y > 0 which intersects the line y = 0 at the points A(‐1, 0) and B(0, 0). For y < 0 by a piecewise smooth curve Γ through A which meets the characteristic of (1) issued from B at the point P and the curve Γ which consists of the portion PB of the characteristic through B. We obtain sufficient conditions for the uniqueness of the solution of the problem L[u] = f, dnu: = k(y)uxdy – uydx|γ0 = = Ψ(s) for a “general” function k(y), when r(x, y) is not necessarily zero and Γ1 is of a more general form then in the papers of V. P. Egorov [6], [7].
Über das Verhalten der Lösungen der Maxwellschen Randwertaufgabe in Gebieten mit Kegelspitzen
Authors: Saranen J. , Leis R. 
Published: March 2007

AbstractWe study the behavior of the solution E of the Maxwell's boundary value problem ∇ × ∇ × E + λE = F, n × E|r = 0 in domains Ω which have conical boundary points. In a neighbourhood K(R) = B(a,R) ∩ Ω of a singular boundary point a the field E is expanded using a theorem of N. Weck. It e.g. turns out that the solution lies in H1(3)(K(R)) if K(R) is convex.
The effect of a thin inclusion of high rigidity in an elastic body
Authors: Caillerie D. , Nedelec J. C. 
Published: March 2007

AbstractWe study an elastic body comprising an inclusion whose thickness is 2ϵ and for which Lamé's constants are λϵ and μ. We are interested in the limit behaviour of the inclusion, when ϵ → 0 and μϵ → ϵ; that is when the inclusion becomes thiner and more rigid. Different behaviours are possible, according to the rate at which μϵ converges to infinity; the limit inclusion may “vanish” if it is not very rigid, but also it may be entirely solid.
Scattering of elastic waves through a heterogeneous medium
Authors: Codegone M. , Leis R. 
Published: March 2007

AbstractIn this paper we study the elastic wave diffraction in R3 through a heterogeneous medium, with periodic structure, which occupies a bounded domain. We show that, as the period tends to zero, the solution tends, in some sense, to the solution corresponding to the diffraction by an obstacle made of the classical “homogenized medium”. An analogous result is also proved for the scattering frequencies and the associated scattering functions.
Discontinuous self‐excited systems: An asymptotic analysis
Authors: Granoff B. , Freedman M. I. , Ludwig D. 
Published: March 2007

AbstractIn this paper we derive a uniformly valid asymptotic approximation of the periodic solution of a self‐excited system given by the differential equation

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and β1,β2, are positive constants. By uniformly valid asymptotic approximation we mean that no secular terms are present. Our procedure makes use of a nonlinear change of independent variable that transforms the problem from one in which the discontinuities are ϵ dependent to one in which the discontinuities are ϵ independent. We obtain an asymptotic approximation up to order ϵ of the periodic solution and an asymptotic approximation up to order ϵ2 of the period. Some comparisons between our asymptotic results and numerically derived results are given. Application of our technique to other examples of self‐excited systems is discussed. The equation

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is investigated in detail.
Außenraumaufgaben in der linearen Elastizitätstheorie
Authors: Leis R. 
Published: March 2007

AbstractThe paper starts with a short survey of the treatment of initial‐boundary‐value problems in temperature‐free linear elasticity with unisotropic media. The main part of the paper is concerned with exterior initial‐boundary‐value problems in thermoelasticity. In this case the underlying differential operator A is no longer selfadjoint. Thus the spectrum of A has to be discussed. In 2.1 it is shown that all λ with Re λ < 0 belong to the resolvent set. In 2.2 the case G = R3 with homogeneous isotropic media is considered. Let Λ be the essential spectrum in this case. In 2.3 Λ depending on the thermic coupling parameter is discussed. 2.4 treats the spectrum of A assuming the medium to be homogeneous and isotrop outside a large ball. In this case Λ is the essential spectrum for A too. Radiation conditions are formulated. Finally 2.5 presents a short treatment of the time dependent case with Laplace‐transformation.
The positivity principle in Boltzmann‐type equations
Authors: Ebersoldt F. , Leis R. 
Published: March 2007

AbstractClassical theorems about matrices with non‐negative elements can be generalized for application in eigenvalue problems of integro‐differential form appearing in the description of stationary particle transport. In this way the existence and unicity of positive L1‐eigensolutions of the m‐quasi‐completely‐continuous transport operator can be deduced. These solutions can be realized by a method of successive approximation.
Elektrodynamik mit Spin
Authors: Petry W. , Leis R. 
Published: March 2007

AbstractAtomistic equations of the electromagnetic field for a particle with spin are derived from a Lagrangian. These equations are consistent with the equations of motion for such a particle. The resulting phenomenological equations are the well‐known equations of Maxwell for the electromagnetic field in matter. The atomistic field equations for a particle with spin and magnetic moment give a dipole field. This result and the corresponding quantum mechanics for a particle with spin are applied to compute the hyperfine structure of the hydrogen atom by perturbation theory.
Zur Abhängigkeit der Lösung von ut + (h(u))x = 0 von ihren Anfangswerten
Authors: Wick J. , Leis R. 
Published: March 2007

AbstractThe above equation has some remarkable properties. In general a global solution exists in a weak sense only, and this solution is not reversible in time. Furthermore it is known, that the solutions for different initial values can coincide for all t ⩾ t0 > 0, and the set of the initial values with this property is convex. Conditions assuring that this set contains only one element are given. This means a weak form of time‐reversibility.As a global solution exists only in the weak sense, the classical question concerning dependence of the solution on the initial values needs some modification. This problem is dealt with in suitable L1‐norms. It is shown, that the L1‐norm of the difference of two weak solutions with respect to the space variable does not increase in time.
Ghosts in tomography – the null space of the radon transform
Authors: Louis A. K. , Törnig W. 
Published: March 2007

AbstractLet ω0, …, ωp−1 ϵ S1 be p distinct directions and let R denote the Radon transform. The functions f with support in the unit disc and with Rf(s, ωj) = 0 for s ϵ [−1, 1] and j = 0, …, p − 1 are given. „Ghosts”︁ are constructed, i.e. functions in the null space that pretend not existent tumors. Methods to avaoid such ghosts are indicated.
On the classical solutions of the initial value problem for the unmodified non‐linear Vlasov equation I general theory
Authors: Horst E. , Neunzert H. 
Published: March 2007

AbstractWe study existence and uniqueness of classical solutions of the initial value problem for Vlasov's equation with an unmodified Newtonian or Coulomb force. We give a necessary and sufficient condition for the existence of a solution on an interval I.
Necessary and sufficient conditions for the existence of shock curves of hyperbolic 1‐conservation laws
Authors: Chang P. H. , van der Vooren A. I. 
Published: March 2007

AbstractWe present three conditions for the existence of shock curves of hyperbolic 1‐conservation laws. If at least two of the conditions hold, then shock curves exist uniquely in the classical sense. We present an example (Example 2, Section 4) for which two of the conditions do not hold and a new kind of singularity occurs. In Remark 1 we present two other conditions similar to the above three conditions which also imply the existence of shock curves.
Local energy decay of solutions of linearized shallow‐water equations
Authors: Geveci T. , Kok B. , Kreiss H.‐O. 
Published: March 2007

AbstractWe prove the local decay of the energy of the solution to a mixed initial boundary value problem for the linearized shallow‐water equations with constant coefficients, where the domain is a half‐plane, a certain dissipative boundary condition is prescribed and the initial data have compact support contained in the open half‐plane.
Condition number of matrices derived from two classes of integral equations
Authors: Christiansen S. , Meister E. 
Published: March 2007

AbstractWe investigate some integral equations, i. a. the so‐called Kupradze functional equations, where the two variables of the kernel belong to two different point sets. An extensive survey of the literature shows the various applications of these equations.By a discretization of the integral equations they are replaced by systems of linear algebraic equations. The condition number of the corresponding matrices is investigated, analytically and numerically. It is thereby quantitatively found in which way the condition of the matrices deteriorates when the two point sets are moved away from each other.
Asymptotic analysis of the linear Boltzmann equation
Authors: Borysiewicz M. , Mika J. , Spiga G. , Neunzert H. 
Published: March 2007

AbstractThe stiff boundary value problem for the time‐independent linear Boltzmann equation is considered in the variational formulation in terms of the coercive symmetric bilinear form. It is shown that it is asymptotically equivalent to the related variational diffusion problem when the stiffness parameter tends to zero. Higher order perturbations are also discussed.
Partial asymptotic stability via limiting equations
Authors: Bondi P. , Fergola P. , Gambardella L. , Tenneriello C. , Hagedorn P. 
Published: March 2007

AbstractIn this paper the limiting equations approach is applied to study the stability properties with respect to a part of the state variables for nonautonomous dynamical systems. Sufficient conditions are given for uniform asymptotic eventual strongly partial stability and for uniform asymptotic partial stability. An application of the results is given.
Solvability of some overdetermined elliptic system in a domain with corners π/n
Authors: Zajaczkowski W. M. , Piskorek A. 
Published: March 2007

AbstractIn the paper we prove the existence and uniqueness of solutions of the overdetermined elliptic system

where b, ω are given functions, in a domain Ω C R3 with corners π/n, n = 2, 3, … The proof is divided on two steps, we construct a solution for the Laplace equation in a dihedral angle π/n, using the method of reflection and we get an estimate in the norms of the Sobolev spaces in some neighbourhood of the edge. In the dihedral angle system (A) reduces to the Dirichlet and Neumann problems for the Laplace equation. In the next step we prove the existence of solutions in the Sobolev spaces Wpl(Ω) using the existence of generalized solutions of (A).
The sommerfeld half‐plane problem revisited I: The solution of a pair of coupled Wiener‐Hopf integral equations
Authors: Heins Albert. E. , Meister E. 
Published: March 2007

AbstractA study and the solution of an extension of the classical Sommerfeld half‐plane problem which leads to a pair of integral equations of the Wiener‐Hopf type is given. The method of solution is function theoretic in character and employs a combination of the ideas of Wiener and Hopf and Carleman.
A discontinuous nonlinear eigenvalue/Free boundary problem
Authors: Alexander Roger , Rabinowitz P. H. 
Published: March 2007

AbstractWe study the problem

(H is the Heaviside unit step function) in spherical domains Ω of arbitrary dimension. When g = 0, there are two branches of radial solutions; for small nonzero g there are solutions near the corresponding radial solution. Moreover, the set where μ = 1 is in all cases an analytic hypersurface.
On the convergence of the multigroup approximations for submultiplying slab media
Authors: Nelson P. , Victory H. D. , Neunzert H. 
Published: March 2007

AbstractThe stability, convergence, and consistency properties of the steady‐state multigroup model are investigated for submultiplying slab media. These concepts are defined in a Banach space setting in which the norm of the angular flux is the collision density integrated over phase space. It is shown that the multigroup approximations are stable and are both consistent with, and convergent to, the transport equation under the conditions that the maximum fluctuations in the total cross section and in the expected number of secondary particles, arising from each energy level, tend to zero as the energy mesh becomes finer. A concluding discussion deals with pathologies of the multigroup approximation for situations in which these fluctuations do not tend to zero as the norms of the energy partitions. The results in this paper complement the time‐dependent results of Belleni‐Morante and Busoni for isotropic slabs and the results of Nelson for steady‐state rod media.
Subsonic irrotational flows in multiply connected domains
Authors: Feistauer Miloslav , Meister E. 
Published: March 2007

AbstractThe solvability of a boundary value problem for a nonlinar elliptic partial differential equation in a plane, bounded, multiply connected domain is investigated. The solution of the problem is equal to unknown constants on inner components of the boundary. We want to determine these constants so that the solution satisfies the so‐called trailing conditions. The results have immediate applications in hydrodynamics.
Spectral properties of certain stiff problems in elasticity and acoustics
Authors: Geymonat G. , Lobo‐Hidalgo M. , Sanchez‐Palencia E. , Roach G. F. 
Published: March 2007

AbstractWe consider two vibration problems containing a small parameter → 0: a) Vibration of an elastic, slightly compressible body, and b) acoustic vibration of a slightly viscous compressible barotropic fluid in a vessel.The asymptotics of eigenvalues for problem a) is studied by using a uniformly convergent expansion of the stiff type. After a re‐scaling of the spectral parameter, the problem b) reduces to an analogous problem, and we prove that, as ε → 0, infinitely many eigenvalues converge to 0 (which is an eigenvalue of infinite multiplicity of the corresponding inviscid acoustic problem).
Spline methods in geodetic approximation problems
Authors: Freeden Willi , Törnig W. 
Published: March 2007

AbstractThis paper is concerned with spline methods in a reproducing kernel Hilbert space consisting of functions defined and harmonic in the outer space of a regular surface (e.g. sphere, ellipsoid, telluroid, geoid, (regularized) earth's surface). Spline methods are used to solve interpolation and smoothing problems with respect to a (fundamental) system of linear functional giving information about earth's gravity field. Best approximations to linear functionals are discussed. The spline of interpolation is characterized as the spline of best approximation in the sense of an appropriate (energy) norm.
Akustische wellen in Gebieten, die von zwei lokal gestorten parallelen ebenen begrenzt sind
Authors: Faulhaber Markus , Werner P. 
Published: March 2007

AbstractWe consider acoustic waves in domains with boundaries, coinciding with two parallel planes outside a sufficiently large sphere. Several results on spectral properties of the Laplace operator in such domains are derived and used to prove uniqueness and existence of a solution of the Dirichlet boundary value problem for the reduced wave equation under additional restrictions. In particular, a class of domains is described for which no eigenvalues are present.
A minimax principle for nonlinear eigenvalue problems with applications to nonoverdamped systems
Authors: Voss H. , Werner B. , Hadeler K. P. 
Published: March 2007

AbstractThe theory of Rayleigh functionals for non‐linear eigenvalue problems T(λ) u = 0 is extended to cases where the functional is defined only on a proper subset. The theory applies to problems which do not satisfy an overdamping condition and yields a minimax characterization of eigenvalues. Applications to damped free vibrations of an elastic body are discussed.
Applications of bloch expansion to periodic elastic and viscoelastic media
Authors: Turbe N. , Wilcox C. 
Published: March 2007

AbstractWe consider an unbounded, elastic non homogeneous material with periodic structure. It is shown that the solution of the equations of motion can be expanded in eigenfunctions of periodic operators. The Bloch expansion is used to prove that when the wave‐length is long compared to the period of the structure (related to a small parameter ϵ), the first term of the exact solution expansion, in powers of ϵ, is the solution of the equations obtained in homogenization theory. The case of a long memory linear viscoelastic material is then studied.