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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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