Softcover ISBN:  9781470465551 
Product Code:  GSM/213.S 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Electronic ISBN:  9781470465544 
Product Code:  GSM/213.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 

Book DetailsGraduate Studies in MathematicsVolume: 213; 2021; 322 ppMSC: Primary 35; Secondary 49; 37;
This book gives an extensive survey of many important topics in the theory of Hamilton–Jacobi equations with particular emphasis on modern approaches and viewpoints. Firstly, the basic wellposedness theory of viscosity solutions for firstorder Hamilton–Jacobi equations is covered. Then, the homogenization theory, a very active research topic since the late 1980s but not covered in any standard textbook, is discussed in depth. Afterwards, dynamical properties of solutions, the Aubry–Mather theory, and weak Kolmogorov–Arnold–Moser (KAM) theory are studied. Both dynamical and PDE approaches are introduced to investigate these theories. Connections between homogenization, dynamical aspects, and the optimal rate of convergence in homogenization theory are given as well.
The book is selfcontained and is useful for a course or for reference. It can also serve as a gentle introductory reference to the homogenization theory.ReadershipGraduate students and researchers interested in Hamilton–Jacobi equations and viscosity solutions.

Table of Contents

Chapters

Introduction to viscosity solutions for Hamilton–Jacobi equations

Firstorder Hamilton–Jacobi equations with convex Hamiltonians

Firstorder Hamilton–Jacobi equations with possibly nonconvex Hamiltonians

Periodic homogenization theory for Hamilton–Jacobi equations

Almost periodic homogenization theory for Hamilton–Jacobi equations

Firstorder convex Hamilton–Jacobi equations in a torus

Introduction to weak KAM theory

Further properties of the effective Hamiltonians in the convex setting

Notations

Sion’s minimax theorem

Characterization of the Legendre transform

Existence and regularity of minimizers for action functionals

Boundary value problems

Supconvolutions

Sketch of proof of Theorem 6.26

Solutions to some exercises


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This book gives an extensive survey of many important topics in the theory of Hamilton–Jacobi equations with particular emphasis on modern approaches and viewpoints. Firstly, the basic wellposedness theory of viscosity solutions for firstorder Hamilton–Jacobi equations is covered. Then, the homogenization theory, a very active research topic since the late 1980s but not covered in any standard textbook, is discussed in depth. Afterwards, dynamical properties of solutions, the Aubry–Mather theory, and weak Kolmogorov–Arnold–Moser (KAM) theory are studied. Both dynamical and PDE approaches are introduced to investigate these theories. Connections between homogenization, dynamical aspects, and the optimal rate of convergence in homogenization theory are given as well.
The book is selfcontained and is useful for a course or for reference. It can also serve as a gentle introductory reference to the homogenization theory.
Graduate students and researchers interested in Hamilton–Jacobi equations and viscosity solutions.

Chapters

Introduction to viscosity solutions for Hamilton–Jacobi equations

Firstorder Hamilton–Jacobi equations with convex Hamiltonians

Firstorder Hamilton–Jacobi equations with possibly nonconvex Hamiltonians

Periodic homogenization theory for Hamilton–Jacobi equations

Almost periodic homogenization theory for Hamilton–Jacobi equations

Firstorder convex Hamilton–Jacobi equations in a torus

Introduction to weak KAM theory

Further properties of the effective Hamiltonians in the convex setting

Notations

Sion’s minimax theorem

Characterization of the Legendre transform

Existence and regularity of minimizers for action functionals

Boundary value problems

Supconvolutions

Sketch of proof of Theorem 6.26

Solutions to some exercises