**Graduate Studies in Mathematics**

Volume: 213;
2021;
322 pp;
Softcover

MSC: Primary 35;
Secondary 49; 37

**Print ISBN: 978-1-4704-6555-1
Product Code: GSM/213.S**

List Price: $85.00

AMS Member Price: $68.00

MAA Member Price: $76.50

**Electronic ISBN: 978-1-4704-6554-4
Product Code: GSM/213.E**

List Price: $85.00

AMS Member Price: $68.00

MAA Member Price: $76.50

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#### Supplemental Materials

# Hamilton-Jacobi Equations: Theory and Applications

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*Hung Vinh Tran*

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.

#### Readership

Graduate students and researchers interested in Hamilton–Jacobi equations and viscosity solutions.

#### Table of Contents

# Table of Contents

## Hamilton-Jacobi Equations: Theory and Applications

- Cover Cover11
- Title page iii4
- Preface xi12
- Chapter 1. Introduction to viscosity solutions for Hamilton–Jacobi equations 116
- 1.1. Introduction 116
- 1.2. Vanishing viscosity method for first-order Hamilton–Jacobi equations 520
- 1.3. Existence of viscosity solutions via the vanishing viscosity method 1227
- 1.4. Consistency and stability of viscosity solutions 1530
- 1.5. The comparison principle and uniqueness result for static problems 1631
- 1.6. The comparison principle and uniqueness result for Cauchy problems 2136
- 1.7. Introduction to the classical Bernstein method 2540
- 1.8. Introduction to Perron’s method 2843
- 1.9. Lipschitz estimates for Cauchy problems using Perron’s method 3449
- 1.10. Finite speed of propagation for Cauchy problems 3651
- 1.11. Rate of convergence of the vanishing viscosity process for static problems via the doubling variables method 3853
- 1.12. Rate of convergence of the vanishing viscosity process for static problems via the nonlinear adjoint method 4257
- 1.13. References 4762

- Chapter 2. First-order Hamilton–Jacobi equations with convex Hamiltonians 5166
- 2.1. Introduction to the optimal control theory 5166
- 2.2. Dynamic Programming Principle 5469
- 2.3. Static Hamilton–Jacobi equation for the value function 5772
- 2.4. Legendre’s transform 6075
- 2.5. The optimal control formula from the Lagrangian viewpoint 6378
- 2.6. A further hidden structure of convex first-order Hamilton–Jacobi equations 7085
- 2.7. Maximal subsolutions and their representation formulas 7792
- 2.8. References 87102

- Chapter 3. First-order Hamilton–Jacobi equations with possibly nonconvex Hamiltonians 89104
- Chapter 4. Periodic homogenization theory for Hamilton–Jacobi equations 109124
- 4.1. Introduction to periodic homogenization theory 109124
- 4.2. Cell problems and periodic homogenization of static Hamilton–Jacobi equations 113128
- 4.3. Periodic homogenization for Cauchy problems 117132
- 4.4. Some first properties of the effective Hamiltonian 121136
- 4.5. Further properties of the effective Hamiltonian in the convex setting 126141
- 4.6. Some representation formulas of the effective Hamiltonian in nonconvex settings 136151
- 4.7. Rates of convergence 152167
- 4.8. Nonuniqueness of solutions to the cell problems 158173
- 4.9. References 161176

- Chapter 5. Almost periodic homogenization theory for Hamilton–Jacobi equations 163178
- 5.1. Introduction to almost periodic homogenization theory 163178
- 5.2. Vanishing discount problems and identification of the effective Hamiltonian 166181
- 5.3. Nonexistence of sublinear correctors 168183
- 5.4. Homogenization for Cauchy problems 170185
- 5.5. Properties of the effective Hamiltonians 172187
- 5.6. References 176191

- Chapter 6. First-order convex Hamilton–Jacobi equations in a torus 177192
- 6.1. New representation formulas for solutions of the discount problems 177192
- 6.2. New representation formula for the effective Hamiltonian and applications 182197
- 6.3. Cell problems, backward characteristics, and applications 188203
- 6.4. Optimal rate of convergence in periodic homogenization theory 194209
- 6.5. Equivalent characterizations of Lipschitz viscosity subsolutions 202217
- 6.6. References 204219

- Chapter 7. Introduction to weak KAM theory 207222
- Chapter 8. Further properties of the effective Hamiltonians in the convex setting 247262
- Appendix A. Notations 261276
- Appendix B. Sion’s minimax theorem 267282
- Appendix C. Characterization of the Legendre transform 271286
- Appendix D. Existence and regularity of minimizers for action functionals 277292
- Appendix E. Boundary value problems 283298
- Appendix F. Sup-convolutions 289304
- Appendix G. Sketch of proof of Theorem 6.26 293308
- Appendix H. Solutions to some exercises 297312
- Bibliography 311326
- Index 319334
- Back Cover Back Cover1339