Lecture 8, regularity for 2nd order ellpitic equations

On November 28, we discussed regularity of weak solutions of second order linear elliptic equations, following the standard scheme:

• There is a necessary ingredient: one needs the characterization of Sobolev spaces via difference quotients;

• Step 1: weak solutions of an elliptic equation of the second order have week solutions (locally) in $W^{2,2}$;

• Step 2: iteration of that and local estimates for higher order derivatives,

• Step 3: for smooth coefficients, weak solutions are in $W^{m,2}$ for all $m\in \mathbb{N}$; thus, via Sobolev and Morrey imbedding theorems, weak solutions must be smooth.

You will find that material in L.C. Evans, Chapter 6.3.