We consider a class of abstract evolution problems characterized by the sum of two unbounded linear operators $A$ and $B$, where $A$ is assumed to generate a positive semigroup of contractions on an $L^1$-space and $B$ is positive. We study the relations between the semigroup generator $G$ and the operator $A + B$. $A$ characterization theorem for $G =A+B$ is stated. The results are based on the spectral analysis of $B(\lambda-A)^{-1}$. The main point is to study the conditions under which the value 1 belongs to the resolvent set, the continuous spectrum, or the residual spectrum of \ud
$B(\lambda - A)^{-1}$
NE: Bandle, Catherine [Hrsg.]; Internationale Tagung über Allgemeine Ungleichungen <7, 1995, Oberwolfach>; GT This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use the permission of the copyright owner must be obtained.
Sobolev inequalities in two-dimensional hyperbolic space I[-][ are dealt with. Here [HI is modeled on the upper Euclidean half-plane equipped with the Poincar6-Bergman metric. Some borderline inequalities, where the leading exponent equals the dimension, are focused. The technique involves rearrangements of functions, and tools from calculus of variations and ordinary differential equations.
SUMMARYWe study an inverse problem for photon transport in a host medium (e.g. an interstellar cloud), that occupies a bounded and strictly convex region ⊂ R 3 . Under the assumption that the cross-sections and the sources are known, we identify the boundary surface = @ (within a suitable family F of surfaces), provided that one value of the photon number density is measured at some given location far from .
Consider an interstellar cloud that occupies the region V ⊂ R 3 , bounded by the known surface ∂V , and assume that the scattering cross section s and the total cross section are also known. Then, we prove that it is possible to identify the source q = q(x,t) that produces UV-photons inside the cloud, provided that the UV-photon distribution function arriving at a location x, far from the cloud, is measured at times t 0 , t 1 = t 0 + , . . . , t J = t 0 + J .
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