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

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

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[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.

Proof

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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

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[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 $k - 1$ .

Proof

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