Global Analysis — Local elliptic regularity

3.1 Local elliptic regularity

The main theorem of elliptic regularity is the following:

Theorem (Local elliptic regularity)

Let LL be a differential operator of order k1k \geq 1 on Rn\mathbb {R}^ n, URnU \subseteq \mathbb {R}^ n precompact and LL elliptic over Uˉ\bar{U}. Then

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

    us+kA(us+Lus). \| u\| _{s + k} \leq A(\| u\| _ s + \| Lu\| _ s).
  2. If uHru \in H^ r for some rr is such that LuHsLu \in H^ s for some ss, then μuHs+k(U)\mu u \in H^{s + k}(U) for every μCc(U)\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 LL has constant coefficients, which is easy since

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

for some polynomial pp.

Next, for arbitrary LL, we show that for any xx0x \in x_0, there is a neighbourhood VUV \subseteq U such that the estimate holds for all uu supported in VV. To do so, let L0L_0 be the differential operator with constant coefficients that agree with LL at x0x_0. We then have to bound (LL0)us\| (L - L_0)u\| _ s on the assumption that the coefficients of LL0L - L_0 can be made arbitrarily small near the support of uu.

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 uu were in Hs+kH^{s + k}, then us+k\| u\| _{s + k} would be very well-behaved. This is an a priori estimate. To show that uu is actually in Hs+kH^{s + k}, we apply appropriate smooth approximations.

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

ϕ0,ϕ  dx=1,ϕ(x)=ϕ(x). \phi \geq 0,\quad \int \phi \; \mathrm{d}x = 1,\quad \phi (-x) = \phi (x).

For ε>0\varepsilon > 0, we define

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

We then define the mollifier

Fε:C(Rn)C(Rn)uϕεu.\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

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

  • FεF_\varepsilon commutes with all differential operators with constant coefficients.

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

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

  • For any uHsu \in H^ s, we have FεuuF_\varepsilon u \to u in HsH^ s.

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

The last point is how we are going to show that uHs+k(U)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 μuHs+k1\mu u \in H^{s + k - 1}. We then bound Fεμus+kA(Fεμus+LFεμus)A(Fεμus+[L,Fε]μus+Fε[L,μ]us+FεμLus)Aμus+A1μus+k1+A2us+k1+A3Lus,\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,μ][L, \mu ] is a differential operator of degree k1k - 1.