CiNii 図書 - Hamilton-Jacobi equations : theory and applications

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

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Hamilton-Jacobi equations : theory and applications

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