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MATHEMATICAL PRELIMINARIES

Eduard Feireisl

in Dynamics of Viscous Compressible Fluids

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198528388
eISBN:
9780191713590
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198528388.003.0002
Subject:
Mathematics, Applied Mathematics

The basic mathematical concepts used in the book are presumed in this chapter. The necessary function spaces framework — Lebesgue spaces, Sobolev spaces, and spaces of continuous functions — are ... More


INTEGRAL PROBLEMS

Andrea Braides

in Gamma-Convergence for Beginners

Published in print:
2002
Published Online:
September 2007
ISBN:
9780198507840
eISBN:
9780191709890
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198507840.003.0003
Subject:
Mathematics, Applied Mathematics

This chapter deals with integral problems. Using examples, it illustrates the need to use weak topologies to obtain compactness and coerciveness conditions. The sufficiency and necessity of the ... More


Sobolev Spaces Wk,p(Ω); A Resolution of the Dirichlet Principle

Ali Taheri

in Function Spaces and Partial Differential Equations: Volume 2: Contemporary Analysis

Published in print:
2015
Published Online:
September 2015
ISBN:
9780198733157
eISBN:
9780191797729
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198733157.003.0015
Subject:
Mathematics, Analysis

This chapter introduces the Sobolev spaces of integer order on domains and discusses their embeddings and trace theorems. Here, the starting point is establishing various approximation results by ... More


Elliptic Boundary Value Problems

Karsten Urban

in Wavelet Methods for Elliptic Partial Differential Equations

Published in print:
2008
Published Online:
May 2009
ISBN:
9780198526056
eISBN:
9780191712340
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198526056.003.0003
Subject:
Mathematics, Applied Mathematics, Mathematical Finance

A model problem is introduced, namely the univariate two-point boundary value problem, both with periodic boundary conditions and homogeneous Dirichlet boundary conditions. The chapter describes the ... More


Function Spaces and Partial Differential Equations: Volume 2: Contemporary Analysis

Ali Taheri

Published in print:
2015
Published Online:
September 2015
ISBN:
9780198733157
eISBN:
9780191797729
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198733157.001.0001
Subject:
Mathematics, Analysis

This book presents a comprehensive treatment of aspects of classical and modern analysis relating to theory of ‘partial differential equations’ and the associated ‘function spaces’. It begins with a ... More


Multi-parameter Singular Integrals II: General Theory

Brian Street

in Multi-parameter Singular Integrals. (AM-189)

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691162515
eISBN:
9781400852758
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691162515.003.0005
Subject:
Mathematics, Analysis

This chapter turns to a general theory which generalizes and unifies all of the examples in the preceding chapters. A main issue is that the first definition from the trichotomy does not generalize ... More


Fourier Transform on S′(Rn); The Hilbert-Sobolev Spaces Hs(Rn)

Ali Taheri

in Function Spaces and Partial Differential Equations: Volume 1 - Classical Analysis

Published in print:
2015
Published Online:
September 2015
ISBN:
9780198733133
eISBN:
9780191797712
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198733133.003.0012
Subject:
Mathematics, Analysis

This chapter starts by taking Fourier transform to the space of tempered distributions. It studies the Laplace, heat, wave and Schrödinger equations in the upper half-space. It also looks at ... More


Convergence of Fourier Series: Dini vs. Dirichlet-Jordon

Ali Taheri

in Function Spaces and Partial Differential Equations: Volume 1 - Classical Analysis

Published in print:
2015
Published Online:
September 2015
ISBN:
9780198733133
eISBN:
9780191797712
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198733133.003.0004
Subject:
Mathematics, Analysis

This chapter presents a discussion of various criterions on pointwise and almost everywhere convergence of Fourier series. Introduction of Hilbert-Sobolev spaces on the circle and it also presents ... More


The analysis of the hypoelliptic Laplacian

Jean-Michel Bismut

in Hypoelliptic Laplacian and Orbital Integrals (AM-177)

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691151298
eISBN:
9781400840571
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691151298.003.0012
Subject:
Mathematics, Geometry / Topology

This chapter constructs a functional analytic machinery that is adapted to the analysis of the hypoelliptic Laplacian ℒA,bX. The analysis of the hypoelliptic Laplacian essentially consists in the ... More


Fourier Integral and Sobolev Space Hs

Fon-Che Liu

in Real Analysis

Published in print:
2016
Published Online:
January 2017
ISBN:
9780198790426
eISBN:
9780191831676
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198790426.003.0007
Subject:
Mathematics, Analysis

The Fourier integral, which is often called a Fourier transform, is based on an idea of J. Fourier for resolving functions into basic harmonics in the treatment of heat conduction. The Fourier ... More


Spectral Theory and Differential Operators

David Edmunds and Des Evans

Published in print:
2018
Published Online:
September 2018
ISBN:
9780198812050
eISBN:
9780191861130
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198812050.001.0001
Subject:
Mathematics, Pure Mathematics

This book gives an account of those parts of the analysis of closed linear operators acting in Banach or Hilbert spaces that are relevant to spectral problems involving differential operators, and ... More


Function Spaces and Partial Differential Equations: Volume 1 - Classical Analysis

Ali Taheri

Published in print:
2015
Published Online:
September 2015
ISBN:
9780198733133
eISBN:
9780191797712
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198733133.001.0001
Subject:
Mathematics, Analysis

This book, which is the first volume of two, presents a comprehensive treatment of aspects of classical and modern analysis relating to theory of ‘partial differential equations’ and the associated ... More


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