Green functions of higher-order differential operators

Estimates on Green functions of second order differential operators with singular coefficients

We investigate the Green’s functions G(x;x′) of some second order differential operators on R with singular coefficients depending only on one coordinate x0. We express the Green’s functions by means of the Brownian motion. Applying probabilistic methods we prove that when x = (0,x) and x′ = (0, x′) (here x0 = 0) lie on the singular hyperplanes then G(0,x; 0, x′) is more regular than the Green’...

Positivity of the Green Functions for Higher Order Ordinary Differential Equations

We consider the equation n X k=0 ak(t)x (n−k)(t) = 0, t ≥ 0, where a0(t) ≡ 1, ak(t) (k = 1, . . . , n) are real bounded functions. Assuming that all the roots of the polynomial zn+a1(t)zn−1+· · ·+an(t) (t ≥ 0) are real, we derive positivity conditions for the Green function for the Cauchy problem. We also establish a lower estimate for the Green function and a comparison theorem for solutions.

Operators of higher order

Motivated by results on interactive proof systems we investigate the computational power of quantifiers applied to well-known complexity classes. In special, we are interested in existential, universal and probabilistic bounded error quantifiers ranging over words and sets of words, i.e. oracles if we think in a Turing machine model. In addition to the standard oracle access mechanism, we also ...

Lp SPECTRAL THEORY OF HIGHER-ORDER ELLIPTIC DIFFERENTIAL OPERATORS

Lieb-thirring Inequalities for Higher Order Differential Operators

We derive Lieb-Thirring inequalities for the Riesz means of eigenvalues of order γ≥3/4 for a fourth order operator in arbitrary dimensions. We also consider some extensions to polyharmonic operators, and to systems of such operators, in dimensions greater than one. For the critical case γ=1−1/(2l) in dimension d=1 with l≥2 we prove the inequality L0l,γ,d < Ll,γ,d , which holds in contrast to cu...