Extensions of Moser-Bangert Theory: Locally Minimal Solutions

ISBN-13: 9780817681166 / Angielski / Twarda / 2011 / 208 str.

Paul H. Rabinowitz; Edward W. Stredulinsky

Opis

This self-contained 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.

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."

Recenzje

From the reviews:

"This book contains a study of the solution set to (PDE), expanding work by Moser and Bangert and previous work by the authors for (AC). ... This is an important piece of work concerning a difficult and deep matter. ... This a very good demonstration of the power of variational methods, showing that they can be modified, extended and combined in order to deal with many different kinds of problems." (Jesús Hernández, Mathematical Reviews, Issue 2012 m)

Spis treści

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

Blurb

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×T^n-1 and R²×T^n-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.