Extensions of Moser–Bangert Theory: Locally Minimal Solutions (Progress in Nonlinear Differential Equations and Their Applications Book 81)

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 __R^n^__ 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 __R^2^__×__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. 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 R n 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 R 2 × 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. Front Matter....Pages i-viii Introduction....Pages 1-6 Front Matter....Pages 7-7 Function Spaces and the First Renormalized Functional....Pages 9-22 The Simplest Heteroclinics....Pages 23-35 Heteroclinics in x 1 and x 2 ....Pages 37-52 More Basic Solutions....Pages 53-62 Front Matter....Pages 63-63 The Simplest Cases....Pages 65-79 The Proof of Theorem 6.8....Pages 81-87 k -Transition Solutions for k > 2....Pages 89-96 Monotone 2-Transition Solutions....Pages 97-118 Monotone Multitransition Solutions....Pages 119-129 A Mixed Case....Pages 131-153 Front Matter....Pages 155-155 A Class of Strictly 1-Monotone Infinite Transition Solutions of (PDE)....Pages 157-177 Solutions of (PDE) with Two Transitions in x 1 and Heteroclinic Behavior in x 2 ....Pages 179-203 Back Matter....Pages 205-208 "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 R[superscript n] and an Allen-Cahn PDE model of phase transitions."--P. [4] of cover. "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 R[superscript n] and an Allen-Cahn PDE model of phase transitions."--Page 4 of cover 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. Pt.1. Basic solutions Pt.2. Shadowing cases Pt.3. Solutions of (PDE) defind on R2 x T[superscript n]−2.