Hamilton-Jacobi Equations
Theory and Applications
Seiten
2021
American Mathematical Society (Verlag)
978-1-4704-6511-7 (ISBN)
American Mathematical Society (Verlag)
978-1-4704-6511-7 (ISBN)
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An extensive survey of many important topics in the theory of Hamilton-Jacobi equations with particular emphasis on modern approaches and viewpoints.
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 well-posedness theory of viscosity solutions for first-order 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 self-contained and is useful for a course or for references. It can also serve as a gentle introductory reference to the homogenization theory.
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 well-posedness theory of viscosity solutions for first-order 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 self-contained and is useful for a course or for references. It can also serve as a gentle introductory reference to the homogenization theory.
Introduction to viscosity solutions for Hamilton-Jacobi equations
First-order Hamilton-Jacobi equations with convex Hamiltonians
First-order Hamilton-Jacobi equations with possibly nonconvex Hamiltonians
Periodic homogenization theory for Hamilton-Jacobi equations
Almost periodic homogenization theory for Hamilton-Jacobi equations
First-order 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
Sup-convolutions
Sketch of proof of Theorem 6.26
Solutions to some exercises
Bibliography
Index
| Erscheint lt. Verlag | 30.10.2021 |
|---|---|
| Reihe/Serie | Graduate Studies in Mathematics |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 178 x 254 mm |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| ISBN-10 | 1-4704-6511-6 / 1470465116 |
| ISBN-13 | 978-1-4704-6511-7 / 9781470465117 |
| Zustand | Neuware |
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