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Boundary Value Problems with Global Projection Conditions

Authors: Xiaochun Liu, Prof. Bert-Wolfgang Schulze

Publisher: Springer International Publishing

Book Series : Operator Theory: Advances and Applications

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book presents boundary value problems for arbitrary elliptic pseudo-differential operators on a smooth compact manifold with boundary. In this regard, every operator admits global projection boundary conditions, giving rise to analogues of Toeplitz operators in subspaces of Sobolev spaces on the boundary associated with pseudo-differential projections. The book describes how these operator classes form algebras, and establishes the concept for Boutet de Monvel’s calculus, as well as for operators on manifolds with edges, including the case of operators without the transmission property. Further, it shows how the calculus contains parametrices of elliptic elements. Lastly, the book describes natural connections to ellipticity of Atiyah-Patodi-Singer type for Dirac and other geometric operators, in particular spectral boundary conditions with Calderón-Seeley projections and the characterization of Cauchy data spaces.

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Table of Contents

Frontmatter

Boundary Value Problems with Global Projection Conditions

Frontmatter

Chapter 1. Pseudo-differential operators

Abstract

We first outline some notation and well-known material on standard pseudodifferential operators. Proofs, as far as they are skipped here, can be found in textbooks on the pseudo-differential calculus.

Xiaochun Liu, Bert-Wolfgang Schulze

Chapter 2. BVPs with the transmission property

Abstract

Let us first give a motivation for the transmission property of a symbol at the boundary. Given a smooth manifold X of dimension n with boundary Y, we can form the double 2X; which is an open manifold obtained by gluing together two copies X_± of X along the common boundary (here we identify X with X₊, the plus side).

Xiaochun Liu, Bert-Wolfgang Schulze

Chapter 3. Shapiro–Lopatinskii ellipticity

Abstract

We now define the notion of Shapiro–Lopatinskii ellipticity (also known as SLellipticity) of boundary conditions for an operator in Boutet de Monvel’s calculus on a smooth manifold X with boundary Y . The results can be found, for instance, in the monograph [34] of Rempel and Schulze, and of course, also in the work [9] of Boutet de Monvel; see also the monograph of Grubb [19]. Therefore, here we only sketch the proofs.

Xiaochun Liu, Bert-Wolfgang Schulze

Chapter 4. Toeplitz boundary value problems

Abstract

In this section we extend the results of Section 1.2 to the case of a compact C^∞ manifold X with boundary Y.

Xiaochun Liu, Bert-Wolfgang Schulze

Chapter 5. Cutting and pasting of elliptic operators, Cauchy data spaces

Abstract

Let M be a smooth closed manifold which is decomposed as

$$M=X_{-}\cup X_{+}$$

where \(X_{\pm}\) are smooth compact manifolds with common boundary \(\partial X_{-}=\partial X_{+}=: Y=X_{-} \cap X_{+}\).

Xiaochun Liu, Bert-Wolfgang Schulze

Edge Operators with Global Projection Conditions

Frontmatter

Chapter 6. The cone algebra

Abstract

A manifold with smooth boundary can be regarded as a manifold with edge, and boundary value problems can be interpreted as specific edge problems. In the edge case the inner normal turns into the axial variable of a cone transverse to the edge, and the boundary symbol calculus is now replaced by a calculus on the respective cone. In order to treat edge problems with global projection conditions, we first outline the calculus of operators on a manifold with conical singularities.

Xiaochun Liu, Bert-Wolfgang Schulze

Chapter 7. The edge algebra

Abstract

Let N be a smooth closed manifold. The wedge \(W := N^\vartriangle \times \Omega\) for any open \(\Omega \subseteq \mathbb{R}^q\) is an example of a manifold with edge Ω.

Xiaochun Liu, Bert-Wolfgang Schulze

Chapter 8. Edge-ellipticity

Abstract

In the preceding section for operators A in the edge calculus we defined a pair

$$\sigma(\mathcal{A})=(\sigma_{\psi}(\mathcal{A}),\sigma_\wedge(\mathcal{A}))$$

of principal symbols.

Xiaochun Liu, Bert-Wolfgang Schulze

Chapter 9. Toeplitz edge problems

Abstract

Let M be a compact manifold with edge Y.

Xiaochun Liu, Bert-Wolfgang Schulze

BVPs without the Transmission Property

Frontmatter

Chapter 10. The edge approach to BVPs

Abstract

BVPs without the transmission property on a manifold M with smooth boundary will be interpreted as specific edge problems. It is evident that such an M is a manifold with edge in the sense of Section 7.1, where now dim N = 0.

Xiaochun Liu, Bert-Wolfgang Schulze

Chapter 11. Boundary ellipticity

Abstract

Recall that the operators A in the calculus of boundary value problems without the transmission property have a pair

$$\sigma(\mathcal{A})=(\sigma_\psi(\mathcal{A}),\sigma_\partial(\mathcal{A}))$$

of principal symbols.

Xiaochun Liu, Bert-Wolfgang Schulze

Chapter 12. Toeplitz boundary value problems without the transmission property

Abstract

Let M be a compact manifold with boundary Y.

Xiaochun Liu, Bert-Wolfgang Schulze

Chapter 13. Examples, applications and remarks

Abstract

The present section gives an abstract on additional results around the nature of cone operators and ellipticity. If proofs are dropped we refer to corresponding material in textbooks or articles.

Xiaochun Liu, Bert-Wolfgang Schulze

Backmatter

Title: Boundary Value Problems with Global Projection Conditions
Authors: Xiaochun Liu
Prof. Bert-Wolfgang Schulze
Copyright Year: 2018
Publisher: Springer International Publishing
Electronic ISBN: 978-3-319-70114-1
Print ISBN: 978-3-319-70113-4
DOI: https://doi.org/10.1007/978-3-319-70114-1

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