This is the first half of the year-long introductory graduate course on PDE. PDE is an enormously vast field, and is an important tool for studying problems in many areas of mathematics, in particular in mathematical physics, geometry, and applied mathematics. We will introduce the most useful methods and techniques through studying some prototype equations ---to build up experience and intuition--- rather than emphasizing the most general theories. This first semester course will be more basic, emphasizing the study of constructions and properties of solutions using relatively elementary tools; but the approaches that we will take will motivate and naturally lead to more sophisticated generalizations in more advanced treatment.
Course notes
A solid background in analysis at the level of first seven chapters of baby Rudin, basics of Fourier series, and the main theorems of vector calculus (e.g., divergence theorem).