Hodge Theory in the Sobolev Topology for the De Rham Complex (Memoirs of the American Mathematical Society)

In this book, the authors treat the full Hodge theory for the de Rham complex when calculated in the Sobolev topology rather than in the $L^2$ topology. The use of the Sobolev topology strikingly alters the problem from the classical setup and gives rise to a new class of elliptic boundary value problems. The study takes place on both the upper half space and on a smoothly bounded domain.

Features:

a good introduction to elliptic theory, pseudo-differential operators, and boundary value problems

theorems completely explained and proved

new geometric tools for differential analysis on domains and manifolds