In this paper we prove that div(|x|β∇u)+|x|αf(u)=0, inB u = 0 on ∂B has infinitely many solutions when f is superlinear and grows subcritically for u ≥ 0 and up to critically for u less than 0 with 10, 13 N+β−2 N+β−2 We make extensive use of Pohozaev identities and phase plane and energy arguments.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1965. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

In this paper results concerning a semilinear elliptic boundary value problem are proven. This problem has five solutions when the range of the derivative of the nonlinearity ƒ includes the first two eigenvalues. The existence and multiplicity or radially symmetric solutions under suitable conditions on the nonlinearity when Ω is a ball in R^N.

We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instance. We provide a selfcontained introduction to (nonlocal) elliptic boundary conditions, boundary regularity of solutions, and index theory. In particular, we simplify and generalize the traditional theory of elliptic boundary value problems for Dirac type operators. We also prove a related decomposition theorem, a general version of Gromov and Lawson's relative index theorem and a generalization of the cobordism theorem.

We construct a theory of general boundary value problems for differential operators whose symbols do not necessarily satisfy the Atiyah-Bott condition [3] of vanishing of the corresponding obstruction. A condition for these problems to be Fredholm is introduced and the corresponding finiteness theorems are proved.

We give the homotopy classification and compute the index of boundary value problems for elliptic equations. The classical case of operators that satisfy the Atiyah-Bott condition is studied first. We also consider the general case of boundary value problems for operators that do not necessarily satisfy the Atiyah-Bott condition.

Systems of grid equations that approximate elliptic boundary value problems on locally modified grids are considered. The triangulation, which approximates the boundary with second order of accuracy, is generated from an initial uniform triangulation by shifting nodes near the boundary according to special rules. This "locally modified" grid possesses several significant features: this triangulation has a regular structure, the generation of the triangulation is rather fast, this construction allows to use multilevel preconditioning (BPX-like) methods. The proposed iterative methods for solving elliptic boundary value problems approximately are based on two approaches: The fictitious space method, i.e. the reduction of the original problem to a problem in an auxiliary (fictitious) space, and the multilevel decomposition method, i.e. the construction of preconditioners by decomposing functions on hierarchical grids. The convergence rate of the corresponding iterative process with the preconditioner obtained is independent of the mesh size. The construction of the grid and the preconditioning operator for the three dimensional problem can be done in the same way.

Nesta tese vamos apresentar dois resultados a respeito de K-teoria de álgebras C^{*} de classes de operadores pseudodiferenciais que são globalmente definidos em \\mathbb^. O primeiro resultado é a prova da regularidade da função \\eta para operadores clássicos com símbolos de Shubin. Vamos mostrar que a álgebra de operadores pseudodiferenciais em \\mathbb^ com símbolos de Shubin permite a construção de potências complexas e um tipo de traço de Kontsevich-Vishik numa forma muito similar àquela feita para variedades compactas, com definições até mais simples. Mostraremos, então, que podemos definir as funções \\zeta e \\eta também para esses símbolos. Finalmente mostraremos como o conhecimento de fatos simples sobre a sua K-teoria permitem a prova da regularidade da função \\eta. Para variedades compactas, esse resultado tem muitas implicações. Acreditamos assim que ele também possa ser interessante para os estudos de operadores globais em \\mathbb^. O segundo resultado é o cálculo da K-teoria de operadores limitados gerados por operadores de Boutet de Monvel SG de ordem (0,0) e tipo zero em \\mathbb_{+}^. Boutet de Monvel introduziu a álgebra que leva o seu nome para estudar o índice de operadores elípticos de fronteira em variedades compactas com bordo. Mais recentemente uma nova abordagem foi proposta por Melo, Nest, Schrohe e Schick para obter resultados sobre o índice de Fredholm usando a K-teoria de álgebras C^{*}, uma ferramenta que não era disponível ainda quando Boutet de Monvel desenvolveu sua álgebra. Nossa ideia foi, então, mostrar como calcular a K-teoria de álgebras de Boutet de Monvel com símbolos SG em \\mathbb_{+}^, em que os símbolos SG são uma classe de símbolos globalmente definidos em \\mathbb^. Acreditamos que isso possa ser útil também ao estudo de problemas elípticos de fronteira para operadores de Boutet de Monvel com símbolos SG em certas classes de variedades não compactas. / We are going to present two results concerning K-theory of C^{*} algebras of classes of pseudodifferential operators that are globally defined in \\mathbb^. The first result is the proof of the regularity of the \\eta function for classical operators with Shubin symbols. We are going to show that the algebra of classical pseudodifferential operators in \\mathbb^ with Shubin symbols allows the construction of complex powers and a kind of Kontsevich-Vishik trace in a very similar way as on compact manifolds, with even easier definitions. Then we show that we can define the \\zeta and \\eta functions also for these symbols. Finally we will show how the knowledge of simple facts about the K-theory of pseudodifferential operators with Shubin\'s symbols allows the proof of the regularity of the \\eta function at 0. For compact manifolds, this regularity is a result that has many implications. Therefore it may also be interesting for global operators in \\mathbb^. The second result is the evaluation of the K-theory of bounded operators generated by SG Boutet de Monvel operators of order (0,0) and type 0 in \\mathbb_^. Boutet de Monvel introduced his algebra to study the index of elliptic boundary value problems on compact manifolds. More recently a new approach was proposed by Melo, Nest, Schrohe and Schick to obtain results about the index of Fredholm operators using the K-theory of C^ algebras, a tool which was not well known when Boutet de Monvel published his work. The idea here is to show how one can evaluate the K-theory of the Boutet de Monvel operators with SG symbols in \\mathbb_^, where SG symbols is a class of symbols globally defined in \\mathbb^. We believe that this can be useful to the study of index of Fredholm problems also in the case of Boutet de Monvel operators with SG symbols in some classes of non-compact manifolds.

Nesta tese vamos apresentar dois resultados a respeito de K-teoria de álgebras C^{*} de classes de operadores pseudodiferenciais que são globalmente definidos em \\mathbb^. O primeiro resultado é a prova da regularidade da função \\eta para operadores clássicos com símbolos de Shubin. Vamos mostrar que a álgebra de operadores pseudodiferenciais em \\mathbb^ com símbolos de Shubin permite a construção de potências complexas e um tipo de traço de Kontsevich-Vishik numa forma muito similar àquela feita para variedades compactas, com definições até mais simples. Mostraremos, então, que podemos definir as funções \\zeta e \\eta também para esses símbolos. Finalmente mostraremos como o conhecimento de fatos simples sobre a sua K-teoria permitem a prova da regularidade da função \\eta. Para variedades compactas, esse resultado tem muitas implicações. Acreditamos assim que ele também possa ser interessante para os estudos de operadores globais em \\mathbb^. O segundo resultado é o cálculo da K-teoria de operadores limitados gerados por operadores de Boutet de Monvel SG de ordem (0,0) e tipo zero em \\mathbb_{+}^. Boutet de Monvel introduziu a álgebra que leva o seu nome para estudar o índice de operadores elípticos de fronteira em variedades compactas com bordo. Mais recentemente uma nova abordagem foi proposta por Melo, Nest, Schrohe e Schick para obter resultados sobre o índice de Fredholm usando a K-teoria de álgebras C^{*}, uma ferramenta que não era disponível ainda quando Boutet de Monvel desenvolveu sua álgebra. Nossa ideia foi, então, mostrar como calcular a K-teoria de álgebras de Boutet de Monvel com símbolos SG em \\mathbb_{+}^, em que os símbolos SG são uma classe de símbolos globalmente definidos em \\mathbb^. Acreditamos que isso possa ser útil também ao estudo de problemas elípticos de fronteira para operadores de Boutet de Monvel com símbolos SG em certas classes de variedades não compactas. / We are going to present two results concerning K-theory of C^{*} algebras of classes of pseudodifferential operators that are globally defined in \\mathbb^. The first result is the proof of the regularity of the \\eta function for classical operators with Shubin symbols. We are going to show that the algebra of classical pseudodifferential operators in \\mathbb^ with Shubin symbols allows the construction of complex powers and a kind of Kontsevich-Vishik trace in a very similar way as on compact manifolds, with even easier definitions. Then we show that we can define the \\zeta and \\eta functions also for these symbols. Finally we will show how the knowledge of simple facts about the K-theory of pseudodifferential operators with Shubin\'s symbols allows the proof of the regularity of the \\eta function at 0. For compact manifolds, this regularity is a result that has many implications. Therefore it may also be interesting for global operators in \\mathbb^. The second result is the evaluation of the K-theory of bounded operators generated by SG Boutet de Monvel operators of order (0,0) and type 0 in \\mathbb_^. Boutet de Monvel introduced his algebra to study the index of elliptic boundary value problems on compact manifolds. More recently a new approach was proposed by Melo, Nest, Schrohe and Schick to obtain results about the index of Fredholm operators using the K-theory of C^ algebras, a tool which was not well known when Boutet de Monvel published his work. The idea here is to show how one can evaluate the K-theory of the Boutet de Monvel operators with SG symbols in \\mathbb_^, where SG symbols is a class of symbols globally defined in \\mathbb^. We believe that this can be useful to the study of index of Fredholm problems also in the case of Boutet de Monvel operators with SG symbols in some classes of non-compact manifolds.

The paper is concerned with the finite element resolution of layers appearing in singularly perturbed problems. A special anisotropic grid of Shishkin type is constructed for reaction diffusion problems. Estimates of the finite element error in the energy norm are derived for two methods, namely the standard Galerkin method and a stabilized Galerkin method. The estimates are uniformly valid with respect to the (small) diffusion parameter. One ingredient is a pointwise description of derivatives of the continuous solution. A numerical example supports the result. Another key ingredient for the error analysis is a refined estimate for (higher) derivatives of the interpolation error. The assumptions on admissible anisotropic finite elements are formulated in terms of geometrical conditions for triangles and tetrahedra. The application of these estimates is not restricted to the special problem considered in this paper.