Umfang:
Online-Ressource (VIII, 208p, digital)
ISBN:
9780817681173
Serie:
Progress in Nonlinear Differential Equations and Their Applications 81
Inhalt:
1 Introduction -- Part I: Basic Solutions -- 2 Function Spaces and the First Renormalized Functional -- 3 The Simplest Heteroclinics -- 4 Heteroclinics in x1 and x2 -- 5 More Basic Solutions -- Part II: Shadowing Results -- 6 The Simplest Cases -- 7 The Proof of Theorem 6.8 -- 8 k-Transition Solutions for k 〉 2 -- 9 Monotone 2-Transition Solutions -- 10 Monotone Multitransition Solutions -- 11 A Mixed Case -- Part III: Solutions of (PDE) Defined on R^2 x T^{n-2} -- 12 A Class of Strictly 1-Monotone Infinite Transition Solutions of (PDE) -- 13 Solutions of (PDE) with Two Transitions in x1 and Heteroclinic Behavior in x2.
Inhalt:
With the goal of establishing a version for partial differential equations (PDEs) of the Aubry–Mather theory of monotone twist maps, Moser and then Bangert studied solutions of their model equations that possessed certain minimality and monotonicity properties. This monograph presents extensions of the Moser–Bangert approach that include solutions of a family of nonlinear elliptic PDEs on Rn and an Allen–Cahn PDE model of phase transitions. After recalling the relevant Moser–Bangert results, Extensions of Moser–Bangert Theory pursues the rich structure of the set of solutions of a simpler model case, expanding upon the studies of Moser and Bangert to include solutions that merely have local minimality properties. Subsequent chapters build upon the introductory results, making the monograph self contained. Part I introduces a variational approach involving a renormalized functional to characterize the basic heteroclinic solutions obtained by Bangert. Following that, Parts II and III employ these basic solutions together with constrained minimization methods to construct multitransition heteroclinic and homoclinic solutions on R×Tn-1 and R2×Tn-2, respectively, as local minima of the renormalized functional. The work is intended for mathematicians who specialize in partial differential equations and may also be used as a text for a graduate topics course in PDEs.
Anmerkung:
Includes bibliographical references and index
,
Extensions of Moser-Bangert Theory; Preface; Contents; Chapter 1 Introduction; Part I Basic Solutions; Chapter 2 Function Spaces and the First Renormalized Functional; Chapter 3 The Simplest Heteroclinics; Chapter 4 Heteroclinics in x1 and x2; Chapter 5 More Basic Solutions; 5.1 Higher-Dimensional Heteroclinics; 5.2 Other Coordinate Systems; 5.3 Generalizations to α E Qn; Part II Shadowing Results; Chapter 6 The Simplest Cases; Chapter 7 The Proof of Theorem 6.8; Chapter 8 k-Transition Solutions for k 〉 2; Chapter 9 Monotone 2-Transition Solutions
,
Chapter 10 Monotone Multitransition SolutionsChapter 11 A Mixed Case; Part III Solutions of (PDE) Defined on R2 x Tn-2; Chapter 12 A Class of Strictly 1-Monotone Infinite Transition Solutions of (PDE); Chapter 13 Solutions of (PDE) with Two Transitions in x1 and Heteroclinic Behavior in x2; References; Index
Weitere Ausg.:
ISBN 9780817681166
Weitere Ausg.:
Erscheint auch als Druck-Ausgabe Rabinowitz, Paul H., 1939 - Extensions of Moser-Bangert theory New York, NY : Birkhäuser, 2011 ISBN 0817681167
Weitere Ausg.:
ISBN 9780817681166
Sprache:
Englisch
Fachgebiete:
Mathematik
Schlagwort(e):
Nichtlineare elliptische Differentialgleichung
;
Minimallösung
;
Nichtlineare elliptische Differentialgleichung
;
Minimallösung
DOI:
10.1007/978-0-8176-8117-3
URL:
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URL:
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