Global Analysis — Local elliptic regularity

## 3.1 Local elliptic regularity

The main theorem of elliptic regularity is the following:

Theorem (Local elliptic regularity)

Let $L$ be a differential operator of order $k \geq 1$ on $\mathbb {R}^ n$, $U \subseteq \mathbb {R}^ n$ precompact and $L$ elliptic over $\bar{U}$. Then

1. There is a constant $A$ such that for all $u \in H^{s + k}(U)$, we have

$\| u\| _{s + k} \leq A(\| u\| _ s + \| Lu\| _ s).$
2. If $u \in H^ r$ for some $r$ is such that $Lu \in H^ s$ for some $s$, then $\mu u \in H^{s + k}(U)$ for every $\mu \in C_ c^\infty (U)$.

The first part requires getting our hands dirty and proving explicit estimates. A proof sketch will be given here, with the details carried out in the Appendix.

Proof sketch of (1)

First prove it if $L$ has constant coefficients, which is easy since

$\widehat{Lu}(\xi ) = p(\xi ) \hat{u}(\xi )$

for some polynomial $p$.

Next, for arbitrary $L$, we show that for any $x \in x_0$, there is a neighbourhood $V \subseteq U$ such that the estimate holds for all $u$ supported in $V$. To do so, let $L_0$ be the differential operator with constant coefficients that agree with $L$ at $x_0$. We then have to bound $\| (L - L_0)u\| _ s$ on the assumption that the coefficients of $L - L_0$ can be made arbitrarily small near the support of $u$.

Finally, use compactness and partitions of unity to glue these results together.

The second part follows from the first formally.

Observe the first part tells us if we already knew that $u$ were in $H^{s + k}$, then $\| u\| _{s + k}$ would be very well-behaved. This is an a priori estimate. To show that $u$ is actually in $H^{s + k}$, we apply appropriate smooth approximations.

Fix a $\phi \in C_ c^\infty (\mathbb {R}^ n)$ such that

$\phi \geq 0,\quad \int \phi \; \mathrm{d}x = 1,\quad \phi (-x) = \phi (x).$

For $\varepsilon > 0$, we define

$\phi _\varepsilon (x) = \frac{1}{\varepsilon ^ n} \phi \left(\frac{x}{\varepsilon }\right).$

We then define the mollifier

\begin{aligned} F_\varepsilon : C^\infty (\mathbb {R}^ n) & \to C^\infty (\mathbb {R}^ n)\\ u & \mapsto \phi _\varepsilon * u. \end{aligned}

The main theorem about mollifiers, which I will not prove, is that

Theorem

$F_\varepsilon$ extends to a bounded operator $H^ s \to H^ s$ with norm $\leq 1$. Moreover

• $F_\varepsilon$ commutes with all differential operators with constant coefficients.

• If $L$ is a differential operator with compact support, then $[F_\varepsilon , L]$ extends to a map $H^ s \to H^{s - k + 1}$ for all $s \in \mathbb {R}$, and has uniformly bounded operator norm.

• For any $u \in H^ s$, we have $F_\varepsilon u \in C^\infty \cap H^ s$.

• For any $u \in H^ s$, we have $F_\varepsilon u \to u$ in $H^ s$.

• If $U \subseteq \mathbb {R}^ n$ is precompact and $u \in H^ r(U)$ for some $r < s$, and $\| F_ t u\| _ s$ is uniformly bounded in $t$, then $u \in H^ s(U)$.

The last point is how we are going to show that $u \in H^{s + k}(U)$, while the others are needed to establish the required bounds.

Proof of (2) from (1)
Inductively, we may assume $\mu u \in H^{s + k - 1}$. We then bound \begin{aligned} \| F_\varepsilon \mu u\| _{s + k} & \leq A(\| F_\varepsilon \mu u\| _ s + \| L F_\varepsilon \mu u\| _ s)\\ & \leq A(\| F_\varepsilon \mu u\| _ s + \| [L, F_\varepsilon ] \mu u\| _ s + \| F_\varepsilon [L, \mu ] u\| _ s + \| F_\varepsilon \mu Lu\| _ s)\\ & \leq A\| \mu u\| _ s + A_1 \| \mu u\| _{s + k - 1} + A_2\| u\| _{s + k - 1} + A_3 \| Lu\| _ s, \end{aligned} where for the third term, we used that $[L, \mu ]$ is a differential operator of degree $k - 1$.