⇤Math diary
May 10, 2023: Elliptic operator theory yoga
Today i spent a few hours helping my friend with their PDE homework, which was to prove the existance and uniqueness for the nonhomogenous laplace equation, using general sobolev space techniques. It was good for me, and gave me practice with dealing with the nitty gritty of elliptic operator theory. Let me explain how the world elliptic operator theory goes round.
For setup, say that $D$ is an order $k$ linear elliptic operator. Order $k$ means there are at most $k$ derivatives, and “elliptic” means that the coeficents of the highest order terms
There are three theorems of note.
- Lax-Milgram theorem:
- This gives you the existence and uniqueness of weak $L^2$ solutions of the PDE $Du = f$, assuming that $\lVert Du \rVert/\lVert u\rVert$ is uniformly bounded away from zero and infinity. It’s essentially an infinite dimensional inverse function theorem, for linear functions.
- Gardings inequality:
- For elliptic $D$ as above, $\lVert u\rVert_{H^k} \leq C\left(\lVert Du\rVert_{L^2} + \lVert u\rVert_{L^2}\right)$
- Here $\lVert u\rVert_{H^k}$ denotes the order $k$ Sobolev norm, which is the sum of $L^2$ norms of all derivatives of $u$ of order $k$ or less. Bounding the $H^k$ norm of a function means that the first $k$ derivatives are under control. Garding’s inequality says that we can control the first $k$ derivatives with only the $L^2$ norm of $u$ and $Du$. Said another way, an elliptic operator encompasses all lower order derivatives.
- Sobolev embedding theorems:
I’ll finish this later maybe…