Volume 30, Issue 3 -

An asymptotic expansion valid for largepositive values of x is constructed for the Kontorovich–Lebedev transform f(x) = 2/π 2 ∫ 0 ∞ s sinh( s π) K is ( x ) F ( s ) ds where K is ( x ) denotes the MacDonald type Bessel function of imaginary order. The paper obtains the modifications necessary to an asymptotic expansion of the above transform obtained by Wong (1981) when the latter is inapplicable because certain so called “moments” of the function F ( s ) sinh( s π) vanish.

We prove regularity results up to the boundary for time independent generalized Maxwell equations on Riemannian manifolds with boundary using the calculus of alternating differential forms. We discuss homogeneous and inhomogeneous boundary data and show ‘polynomially weighted regularity in exterior domains as well’.

The gradient flow of the surface functional of compact hypersurfaces with respect to the inner product of L 2 is described by the evolution of surfaces by their mean curvature. This type of flows have been studied using methods from differential geometry and the theory of partial differential equations as well. In this paper we are interested in the evolution of surfaces obtained by rotating the graph of a positive and periodic function around the abscissa. The main result of this work completes the studies presented in [2] and [5]. More precisely, we prove that for periodic surfaces with non-negative mean curvature satisfying a natural monotonicity property the solution to the mean curvature flow blows up in finite time in the sense that neck pinching occurs.

Zusammenfassung Wenn der im Banachraum Z dicht definierte Operator A eine stark stetige Halbgruppe { T ( t )} t ≥ 0 in Z erzeugt, so verlangt die Literatur bisher für die Relation: v ( t )≔∫ 0 t T ( t - s ) g ( s ) ds ∈ D ( A ), daß die Funktion g ( s ):[0, T ]→ Z zumindest lipschitzstetig ist (s. dazu z.B. Goldstein [2], Seite 84). D ( A ) ist dabei der Definitionsbereich. In unserer Arbeit beweisen wir u. a.: u ( t )≔∫ t 1 t e i C ( t - s ) F ( s ) ds ∈ D ( C ), u 1 ( t )≔∫ t 1 t cos B ( t - s ) f ( s ) ds ∈ D ( B ) und u 2 ( t )≔∫ t 1 t sin B ( t - s ) f ( s ) ds ∈ D ( B ), wenn die Funktionen F ( s ):[ t 1 , t 2 ]→ X und f ( s ):[ t 1 , t 2 ]→ H lediglich von beschränkter Variation sind. Dabei sind C ein selbstadjungierter Operator im komplexen Hilbertraum X und B ein positiv-definiter, selbstadjungierter Operator im komplexen Hilbertraum H .

Some problems in convexity theory lead to integral equations, generalizing Radon transform on the sphere. We propose a consistency method for the solution of these integral equations. Using this method we find an explicit inversion formula, yielding a practical algorithm to solve the considered equations.

We consider the energy Laplacian Δ ν on the Sierpinski gasket (SG), which is defined by the standard self-similar energy E and the Kusuoka measure, and is different from the standard self-similar Laplacian Δ. We study the local behavior of function u ∈domΔ ν k near a boundary point q 0 . We define jets of local derivatives at q 0 and estimate the decay rate of u near q 0 in terms of the vanishing of jet. This can be used to define Taylor approximating polynomials with error estimates. Analogous results are known for the standard Laplacian, but the results here are quite different. We also confirm experimentally the absolute continuity of different energy measures, and give experimental evidence that the density is p -integrable for p < log(15)/log(9).

Similar as in the classical case of polynomials, as is known, para-orthogonal rational functions on the unit circle can be used to obtain quadrature formulas of Szegő-type to approximate some integrals. In the present paper we carry out a thorough discussion of the existence of such rational functions in terms of the underlying Borel measure on the unit circle.