Convexity of trace functionals and Schrödinger operators

Let H be a semi-bounded self-adjoint operator on a separable Hilbert space. For a certain class of positive, continuous, decreasing, and convex functions F we show the convexity of trace functionals of the form tr(F(H+U−ε(U)))−ε(U), where U is a bounded, self-adjoint operator and ε(U) is a normalizing real function—the Fermi level—which may be identical zero. If additionally F is continuously differentiable, then the corresponding trace functional is Fréchet differentiable and there is an expression of its gradient in terms of the derivative of F. The proof of the differentiability of the trace functional is based upon Birman and Solomyak's theory of double Stieltjes operator integrals. If, in particular, H is a Schrödinger-type operator and U a real-valued function, then the gradient of the trace functional is the quantum mechanical expression of the particle density with respect to an equilibrium distribution function f=−F^′. Thus, the monotonicity of the particle density in its dependence on the potential U of Schrödinger's operator—which has been understood since the late 1980s—follows as a special case.     

Positivity for Perturbations of Polyharmonic Operators with Dirichlet Boundary Conditions in Two Dimensions

Higher order elliptic partial differential equations with Dirichlet boundary conditions in general do not satisfy a maximum principle. Polyharmonic operators on balls are an exception. Here it is shown that in IR² small perturbations of polyharmonic operators and of the domain preserve the maximum principle. Hence the Green function for the clamped plate equation on an ellipse with small eccentricity is positive.     

Global solutions for superlinear parabolic equations involving the biharmonic operator for initial data with optimal slow decay

We are interested in stability/instability of the zero steady state of the superlinear parabolic equation u _t + Δ² u = |u| ^p-1 u in , where the exponent is considered in the “super-Fujita” range p > 1 + 4/n. We determine the corresponding limiting growth at infinity for the initial data giving rise to global bounded solutions. In the supercritical case p > (n + 4)/(n−4) this is related to the asymptotic behaviour of positive steady states, which the authors have recently studied. Moreover, it is shown that the solutions found for the parabolic problem decay to 0 at rate t ^−1/(p-1).     

Supercritical biharmonic equations with power-type nonlinearity

We study two different versions of a supercritical biharmonic equation with a power-type nonlinearity. First, we focus on the equation Δ² u = |u| ^p-1 u over the whole space , where n > 4 and p > (n + 4)/(n − 4). Assuming that p < p _c, where p _c is a further critical exponent, we show that all regular radial solutions oscillate around an explicit singular radial solution. As it was already known, on the other hand, no such oscillations occur in the remaining case p ≥ p _c. We also study the Dirichlet problem for the equation Δ² u = λ (1 + u) ^p over the unit ball in , where λ > 0 is an eigenvalue parameter, while n > 4 and p > (n + 4)/(n − 4) as before. When it comes to the extremal solution associated to this eigenvalue problem, we show that it is regular as long as p < p _c. Finally, we show that a singular solution exists for some appropriate λ > 0.      

Carbonylniobium chemistry: II. Penta- and tetra-carbonylphosphineniobates(I)

A number of symmetrical and unsymmetrical bis-arene-ruthenium cations has been prepared and their reduction with sodium borohydride studied. Hydride hydrogen is shown to add preferentially to the less alkylated ring. The conditions are established, which allow the preparation of a new, previously unknown, cationic complex of arene-cyclohexadienyl-ruthenium by stepwise addition of hydride hydrogen.     

Hardy inequalities with optimal constants and remainder terms

We show that the classical Hardy inequalities with optimal constants in the Sobolev spaces and in higher-order Sobolev spaces on a bounded domain can be refined by adding remainder terms which involve norms. In the higher-order case further norms with lower-order singular weights arise. The case being more involved requires a different technique and is developed only in the space .

     

Study of the biomechanical behaviour of structurally stable/unstable motion segments of the sheep spine

The effects of damage of intervertebral discs on their biomechanical behaviour and the factors favouring the progression of instability are studied. Healthy and damaged movement segments are analyzed experimentally and numerically. The aim is to represent and predict the effects of tissue damage and changes in the spine by comparison with healthy segments. Since the intervertebral disc acts as a mechanical damper, relaxation tests are performed in addition to pressure experiments. The experiments are carried out in a bioreactor with tempered nutrient solution. A cultivation period in the bioreactor allows detecting cell viability, solute diffusion rates and gene expression of the discs. Numerically, the nonlinear, viscoelastic, anisotropic and diffusion-dependent behaviour of the intervertebral disc is modelled with the FE-program Abaqus, using a modular material law as a UMAT subroutine. With the measurement results, the relevant parameters can be determined so that the mechanical behaviour of intervertebral discs can be simulated. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)