Introduction to Modern Partial Differential Equations

Author: Qi Minyou
Publisher:
Publish Date: 2005-10-13
Features: Microlocal analysis, since its inception in the mid-1960s, has made significant progress in advancing the theory of partial differential equations. By the late 1970s, it had become well-established, earning the nickname "the 1970s algorithm." Subsequently, it has further developed in precision and expanded from the linear domain to the nonlinear domain. This clearly represents a promising research direction in the 1990s. The purpose of this book is to introduce these developments concerning two specialized topics: nonlinear singularity analysis and subelliptic problems, many of which are the author's own research achievements. The structure of the book is generally as follows: Chapters 2, 3, and 4 focus on nonlinear microlocal analysis, including the theory of pseudo-differential operators established by J.-M. Bony and nonlinear singularity analysis. The last three chapters cover the theory of pseudo-differential operators on non-homogeneous Sobolev spaces, its applications to subelliptic problems, and the theory of higher-order microlocal analysis, among other topics. Both of these parts are currently active and rapidly developing research fields. To help readers understand the origins of these advancements and facilitate their reading, Chapter provides a systematic and comprehensive introduction to classical microlocal analysis.