An introduction to the geometrical analysis of vector fields : with applications to maximum principles and Lie groups

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An introduction to the geometrical analysis of vector fields : with applications to maximum principles and Lie groups / Stefano Biagi, Andrea Bonfiglioli

資料種別:

図書

出版情報:

New Jersey : World Scientific, c2019

形態:

xxv, 423 p. ; 25 cm

著者名:

ISBN:

9789813276611 [9813276614]

書誌ID:

BB27608045

子書誌情報

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所蔵情報

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書誌詳細

注記:

Includes bibliographical references (p. 415-419) and index

主題:

Lie groups; Maximum principles (Mathematics); Vector fields

言語:

英語

目次情報:

Preface

Flows of Vector Fields in Space / 1：

Notations for vector fields in space / 1.1：

The flow of a vector field / 1.2：

The semigroup property / 1.2.1：

Global vector fields / 1.2.2：

Regular and singular points / 1.2.3：

Differentiation along a flow / 1.3：

The equation of variation for the flow / 1.4：

A Liouville Theorem for ODEs / 1.4.1：

Further regularity of the flow / 1.4.2：

Flowing through X, Y, -X, -Y: commutators / 1.5：

The product of exponentials: motivations / 1.6：

Exercises / 1.7：

The Exponential Theorem / 2：

Main algebraic setting / 2.1：

The Exponential Theorem for K [t] / 2.2：

Two crucial lemmas of non-commutative algebra / 2.2.1：

Poincare's ODE in the formal power series setting / 2.2.2：

The Exponential Theorem for K<> / 2.3：

Dynkin's Formula / 2.4：

A Dynkin-type formula / 2.4.1：

Dynkin's original formula / 2.4.2：

Identities from the Exponential Theorem / 2.5：

The Exponential Theorem for K [s, t] / 2.6：

The algebra K<(x, y> [s, t] / 2.6.1：

The Exponential Theorem for K[s, t] / 2.6.2：

PoincarÃ©'s PDEs on K [s, t] / 2.6.3：

More identities / 2.7：

Appendix: manipulations of formal series / 2.8：

The Composition of Flows of Vector Fields / 2.9：

Again on commutators / 3.1：

Composition of flows of vector fields / 3.2：

Approximation for higher order commutators / 3.3：

Appendix: another identity between formal power series / 3.4：

Hadamard's Theorem for Flows / 3.5：

Preliminaries on derivations and differentials / 4.1：

Time-dependent vector fields / 4.1.1：

Relatedness of vector fields and flows / 4.2：

Invariance of a vector field under a map / 4.2.1：

Commutators and Lie-derivatives / 4.3：

Hadamard's Theorem for flows / 4.4：

Commuting vector fields / 4.5：

Hadamard's Theorem for flows in space / 4.6：

Series expansibility / 4.6.1：

Conjugation of flows / 4.6.2：

The CBHD Operation on Finite Dimensional Lie Algebras / 4.7：

Local convergence of the CBHD series / 5.1：

Recursive identities for Dynkin's polynomials / 5.2：

PoincarÃ©'s ODE on Lie algebras / 5.3：

More Poincare-type ODEs / 5.3.1：

The local associativity of the CBHD series / 5.4：

Appendix: multiple series in Banach spaces / 5.5：

The Connectivity Theorem / 5.6：

HÃ¶rmander systems of vector fields / 6.1：

A useful Linear Algebra lemma / 6.2：

X-subunit curves and X-connectedness / 6.3：

Connectivity for HÃ¶rmander vector fields / 6.3.2：

The X-control distance / 7.1：

Some equivalent definitions of d-x / 7.2：

Basic topological properties of the CC-distance / 7.3：

Euclidean boundedness of the dx balls / 7.3.1：

Length space property / 7.3.2：

The Weak Maximum Principle / 7.4：

Main definitions / 8.1：

Picone's Weak Maximum Principle / 8.2：

Existence of L-barriers / 8.3：

The parabolic Weak Maximum Principle / 8.4：

Appendix: semiellipticity and the WMP / 8.5：

Corollaries of the Weak Maximum Principle / 8.6：

Comparison principles / 9.1：

Maximum-modulus and Maximum Principle / 9.2：

The parabolic case / 9.2.1：

An a priori estimate / 9.3：

Application: Green and Poisson operators / 9.4：

Appendix: Another Maximum Principle / 9.5：

The Maximum Propagation Principle / 9.6：

Assumptions on the operators / 10.1：

Principal vector fields / 10.2：

Propagation and Strong Maximum Principle / 10.3：

Invariant sets and the Nagumo-Bony Theorem / 10.4：

The Hopf Lemma / 10.5：

The proof of the Propagation Principle / 10.6：

The Maximum Propagation along the Drift / 10.7：

Propagation along the drift / 11.1：

The point of view of reachable sets / 11.3：

Examples of propagation sets for a PDO / 11.3.1：

The Differential of the Flow wrt its Parameters / 11.4：

The non-autonomous equation of variation / 12.1：

The autonomous equation of variation / 12.1.1：

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