It is shown that bounded solutions to semilinear elliptic Fuchsian equations obey complete asymptoic expansions in terms of powers and logarithms in the distance to the boundary. For that purpose, Schuze's notion of asymptotic type for conormal asymptotics close to a conical point is refined. This in turn allows to perform explicit calculations on asymptotic types - modulo the resolution of the spectral problem for determining the singular exponents in the asmptotic expansions.

When studyind elliptic operators on manifolds with nonisolated singularities one naturally encounters families of conormal symbols (i.e. operators elliptic with parameter p ∈ IR in the sense of Agranovich-Vishik) parametrized by the set of singular points. For homotopies of such families we define the notion of spectral flow, which in this case is an element of the K-group of the parameter space. We prove that the spectral flow is equal to the index of some family of operators on the infinite cone.

Made available in DSpace on 2015-05-15T11:46:06Z (GMT). No. of bitstreams: 1 ArquivoTotalReginaldo.pdf: 2356428 bytes, checksum: ddfd549c0749b44b5c772848742e7dce (MD5) Previous issue date: 2013-05-17 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work, our main objective is to introduce and investigate certain properties of the so-called primitive ideals of Pellikaan-Siersma, including a version relative to a pair of ideals and a generalization to higher order due to Jiang-Simis, as well as to apply such theory to the study of the conormal module M of an ideal in a quotient ring, with focus on an adequate description of its torsion part T(M) and on the freeness of the torsion-free module M=T (M). The connection between M and the second symbolic power of a certain ideal (the ideal whose conormal module is M) will also be highlighted. / Neste trabalho, nosso principal objetivo é introduzir e investigar certas propriedades dos chamados ideais primitivos de Pellikaan-Siersma, incluindo uma versão relativa a um par de ideais e uma generalização em ordem superior devida a Jiang-Simis, bem como aplicar tal teoria ao estudo do módulo conormal M de um ideal em um anel quociente, com foco em uma descrição adequada de sua torção T(M) e na liberdade do módulo livre-de-torção M=T (M). A conexão entre M e a segunda potência simbólica de um certo ideal (o ideal cujo módulo conormal é M) também será destacada.

Crack problems are regarded as elements in a pseudo-differential algbra, where the two sdes int S± of the crack S are treated as interior boundaries and the boundary Y of the crack as an edge singularity. We employ the pseudo-differential calculus of boundary value problems with the transmission property near int S± and the edge pseudo-differential calculus (in a variant with Douglis-Nirenberg orders) to construct parametrices od elliptic crack problems (with extra trace and potential conditions along Y) and to characterise asymptotics of solutions near Y (expressed in the framework of continuous asymptotics). Our operator algebra with boundary and edge symbols contains new weight and order conventions that are necessary also for the more general calculus on manifolds with boundary and edges.

We study the index problem for families of elliptic operators on manifolds with conical singularities. The relative index theorem concerning changes of the weight line is obtained. AN index theorem for families whose conormal symbols satisfy some symmetry conditions is derived.

Exterior tensor products of elliptic operators on smooth manifolds and manifolds with conical singularities are used to obtain examples of elliptic operators on manifolds with edges that do not admit well-posed edge boundary and coboundary conditions.

We construct elliptic elements in the algebra of (classical pseudo-differential) operators on a manifold M with conical singularities. The ellipticity of any such operator A refers to a pair of principal symbols (σ0, σ1) where σ0 is the standard (degenerate) homogeneous principal symbol, and σ1 is the so-called conormal symbol, depending on the complex Mellin covariable z. The conormal symbol, responsible for the conical singularity, is operator-valued and acts in Sobolev spaces on the base X of the cone. The σ1-ellipticity is a bijectivity condition for all z of real part (n + 1)/2 − γ, n = dimX, for some weight γ. In general, we have to rule out a discrete set of exceptional weights that depends on A. We show that for every operator A which is elliptic with respect to σ0, and for any real weight γ there is a smoothing Mellin operator F in the cone algebra such that A + F is elliptic including σ1. Moreover, we apply the results to ellipticity and index of (operator-valued) edge symbols from the calculus on manifolds with edges.