We study mixed boundary value problems for an elliptic operator A on a manifold X with boundary Y / i.e., Au = f in int X, T±u = g± on int Y±, where Y is subdivided into subsets Y± with an interface Z and boundary conditions T± on Y± that are Shapiro-Lopatinskij elliptic up to Z from the respective sides. We assume that Z ⊂ Y is a manifold with conical singularity v. As an example we consider the Zaremba problem, where A is the Laplacian and T− Dirichlet, T+ Neumann conditions. The problem is treated as a corner boundary value problem near v which is the new point and the main difficulty in this paper. Outside v the problem belongs to the edge calculus as is shown in [3]. With a mixed problem we associate Fredholm operators in weighted corner Sobolev spaces with double weights, under suitable edge conditions along Z {v} of trace and potential type. We construct parametrices within the calculus and establish the regularity of solutions.

We study (pseudo-)differential operators on a manifold with edge Z, locally modelled on a wedge with model cone that has itself a base manifold W with smooth edge Y . The typical operators A are corner degenerate in a specific way. They are described (modulo ‘lower order terms’) by a principal symbolic hierarchy σ(A) = (σ ψ(A), σ ^(A), σ ^(A)), where σ ψ is the interior symbol and σ ^(A)(y, η), (y, η) 2 T*Y 0, the (operator-valued) edge symbol of ‘first generation’, cf. [15]. The novelty here is the edge symbol σ^ of ‘second generation’, parametrised by (z, Ϛ) 2 T*Z 0, acting on weighted Sobolev spaces on the infinite cone with base W. Since such a cone has edges with exit to infinity, the calculus has the problem to understand the behaviour of operators on a manifold of that kind. We show the continuity of corner-degenerate operators in weighted edge Sobolev spaces, and we investigate the ellipticity of edge symbols of second generation. Starting from parameter-dependent elliptic families of edge operators of first generation, we obtain the Fredholm property of higher edge symbols on the corresponding singular infinite model cone.

Operators on a manifold with (geometric) singularities are degenerate in a natural way. They have a principal symbolic structure with contributions from the different strata of the configuration. We study the calculus of such operators on the level of edge symbols of second generation, based on specific quantizations of the corner-degenerate interior symbols, and show that this structure is preserved under compositions.

This thesis will present new results involving Hardy and Hardy-Sobolev-Maz'ya inequalities for fractional integrals. There are two key ingredients to many of these results. The first is the conformal transformation between the upper halfspace and the unit ball. The second is the pseudosymmetric halfspace rearrangement, which is a type of rearrangment on the upper halfspace based on Carlen and Loss' concept of competing symmetries along with certain geometric considerations from the conformal transformation. After reducing to one dimension, we can use the conformal transformation to prove a sharp Hardy inequality for general domains, as well as an improved fractional Hardy inequality over convex domains. Most importantly, the sharp constant is the same as that for the halfspace. Two new Hardy-Sobolev-Maz'ya inequalities will also be established. The first will be a weighted inequality that has a strong relationship with the pseudosymmetric halfspace rearrangement. Then, the psuedosymmetric halfspace rearrangement will play a key part in proving the existence of the standard Hardy-Sobolev-Maz'ya inequality on the halfspace, as well as some results involving the existence of minimizers for that inequality.

This thesis is devoted to the study of mixed norm spaces that arise in connection with embeddings of Sobolev and Besov type spaces. We study different structural, integrability, and smoothness properties of functions satisfying certain mixed norm conditions. Conditions of this type are determined by the behaviour of linear sections of functions. The work in this direction originates in a paper due to Gagliardo (1958), and was further developed by Fournier (1988), by Blei and Fournier (1989), and by Kolyada (2005). Here we continue these studies. We obtain some refinements of known embeddings for certain mixed norm spaces introduced by Gagliardo, and we study general properties of these spaces. In connection with these results, we consider a scale of intermediate mixed norm spaces, and prove intrinsic embeddings in this scale. We also consider more general, fully anisotropic, mixed norm spaces. Our main theorem states an embedding of these spaces to Lorentz spaces. Applying this result, we obtain sharp embedding theorems for anisotropic Sobolev-Besov spaces, and anisotropic fractional Sobolev spaces. The methods used are based on non-increasing rearrangements, and on estimates of sections of functions and sections of sets. We also study limiting relations between embeddings of spaces of different type. More exactly, mixed norm estimates enable us to get embedding constants with sharp asymptotic behaviour. This gives an extension of the results obtained for isotropic Besov spaces by Bourgain, Brezis, and Mironescu, and for anisotropic Besov spaces by Kolyada. We study also some basic properties (in particular the approximation properties) of special weak type spaces that play an important role in the construction of mixed norm spaces, and in the description of Sobolev type embeddings. In the last chapter, we study mixed norm spaces consisting of functions that have smooth sections. We prove embeddings of these spaces to Lorentz spaces. From this result, known properties of Sobolev-Liouville spaces follow.

Thesis (Ph.D.)--McMaster University (Canada), 1995. / Source: Dissertation Abstracts International, Volume: 57-03, Section: B, page: 1839. Adviser: A. Peirce.

Thesis (Ph. D.)--University of Washington, 1999. / Vita. Includes bibliographical references (leaves [86]-88).

Thesis (M.S.)--Brigham Young University. Dept. of Electrical and Computer Engineering, 2004. / Includes bibliographical references (p. 95-99).

Orientador: Sergio Antonio Tozoni / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-06T13:07:08Z (GMT). No. of bitstreams: 1 Oliveira_AndrielberdaSilva_M.pdf: 1284065 bytes, checksum: 0117263cf98921db674e49f5f57d460d (MD5) Previous issue date: 2006 / Resumo: O objetivo da dissertação é realizar um estudo dos espaços de Besov sobre a esfera unitária d-dimensional real Sd. No primeiro capítulo são estudados espaços de interpolação utilizando dois métodos de interpolação real. Em particular são estudados os Teoremas de Equivalência e de Reiteração para os J-método e K-método. No segundo capítulo é realizado um estudo rápido sobre análise harmônica na esfera Sd, incluindo um estudo sobre harmônicos esféricos, harmônicos zonais, somas de Cesàro e sobre um teorema de multiplicadores. O terceiro e último capítulo é o mais importante e nele são aplicados os resultados dos capítulos anteriores. São introduzidos os espaços de Besov, decompondo uma função suave definida sobre a esfera d-dimensional, em uma série de harmônicos esféricos e usando uma seqüência de polinômios zonais que podem ser vistos como uma generalização natural dos polinômios de Vallée Poussin definidos sobre o círculo unitário. O principal resultado estudado diz que todo espaço de Besov pode ser obtido como espaço de interpolação de dois espaços de Sobolev / Abstract: The purpose of this work is to make a study about Besov¿s spaces on the unit d-dimensional real sphere Sd. In the first chapter are studied spaces of interpolation using two real interpolation methods. In particular, are studied The Equivalence Theorem and The Reiteration Theorem for the J-method and the K-method. In the second chapter it is made a short study about harmonic analysis on the sphere Sd, including a study about spherics harmonics, zonal harmonics, Cesàro sums and about a multiplier theorem. The third and last chapter is the most important of this work. In this chapter are applied the results of the others chapters. Are introduced the Besov spaces, decomposing a smooth function defined on the d-dimensional sphere, in a series of harmonics spherics and using a sequence o zonal polynomials which can be seen as a natural generalization of the Vallée Poussin polynomials defined on the unit circle. The main result studied says that every Besov¿s space can be got as a interpolation space of two Sobolev¿s spaces / Mestrado / Mestre em Matemática

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2014-11-25T14:36:31Z No. of bitstreams: 2 Tese - Marcos Leandro Mendes Carvalho - 2013.pdf: 2450216 bytes, checksum: 78d3d3298d2050e0e82310644ecda305 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2014-11-25T14:39:40Z (GMT) No. of bitstreams: 2 Tese - Marcos Leandro Mendes Carvalho - 2013.pdf: 2450216 bytes, checksum: 78d3d3298d2050e0e82310644ecda305 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2014-11-25T14:39:40Z (GMT). No. of bitstreams: 2 Tese - Marcos Leandro Mendes Carvalho - 2013.pdf: 2450216 bytes, checksum: 78d3d3298d2050e0e82310644ecda305 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2013-09-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we develop arguments on the critical point theory for locally Lipschitz functionals on Orlicz-Sobolev spaces, along with convexity, minimization and compactness techniques to investigate existence of solution of the multivalued equation −∆Φu ∈ ∂ j(.,u) +λh in Ω, where Ω ⊂ RN is a bounded domain with boundary smooth ∂Ω, Φ : R → [0,∞) is a suitable N-function, ∆Φ is the corresponding Φ−Laplacian, λ > 0 is a parameter, h : Ω → R is a measurable and ∂ j(.,u) is a Clarke’s Generalized Gradient of a function u %→ j(x,u), a.e. x ∈ Ω, associated with critical growth. Regularity of the solutions is investigated, as well. / Neste trabalho desenvolvemos argumentos sobre a teoria de pontos críticos para funcionais Localmente Lipschitz em Espaços de Orlicz-Sobolev, juntamente com técnicas de convexidade, minimização e compacidade para investigar a existencia de solução da equação multivalente −∆Φu ∈ ∂ j(.,u) +λh em Ω, onde Ω ⊂ RN é um domínio limitado com fronteira ∂Ω regular, Φ : R → [0,∞) é uma N-função apropriada, ∆Φ é o correspondente Φ−Laplaciano, λ > 0 é um parâmetro, h : Ω → R é uma função mensurável e ∂ j(.,u) é o gradiente generalizado de Clarke da função u %→ j(x,u), q.t.p. x ∈ Ω, associada com o crescimento crítico. A regularidade de solução também será investigada.