This is a note for PDE, especially the start of modern PDE theory. The note mainly concentrate on Sobolev spaces(both the classical and the fractional dimensional), linear elliptic, parabolic and wave equations. Why do I say it’s just the start? Because most of conclusions I mentioned in the note are not surprised from the behaviors of the classical solutions. It is no exaggeration to say that we can just keep the original linear equations in our mind while considering these equtions with a little more complicated forms, since we just assume symmetric matrices for simplification and those lower terms are just not essential.

“PDE” by Evans and “Fourier Analysis and Nonlinear PDE” by Bahouri etc. are good references for these contents.