Lemma 10.2.4.E30254F3E614 Let $U\subset \mathbb R^{2n}$ be a bounded open set and let $N\geq 1$. For an open set $W\subset \mathbb R^{2n}$ and an integer $k\geq 0$ let $H^ k(W)$ be the space of $\mathbb C^ N$-valued functions on $W$ whose distributional derivatives of order at most $k$ are square integrable on $W$, with the norm $\| v\| _{(k),W}$ of lemma 10.2.1. For $v\in H^1(U)$ and $\mathbb C^ N$-valued $w\in C^\infty _ c(U)$ put
\[ \mathcal A(v,w)=\int _ U\Bigl (\sum a^{ij}_{IJ}\, \partial _ jv_ I\, \overline{\partial _ iw_ J} +\sum b^ j_{IJ}\, \partial _ jv_ I\, \overline{w_ J}+\sum c_{IJ}\, v_ I\, \overline{w_ J}\Bigr )\mathrm dx, \]
where the sums run over $1\leq i,j\leq 2n$ and $1\leq I,J\leq N$, the coefficients are smooth on a neighborhood of $\overline U$, and the principal coefficients satisfy, for some $\lambda >0$,
\[ \sum a^{ij}_{IJ}(x)\, \zeta _{jI}\, \overline{\zeta _{iJ}}\geq \lambda |\zeta |^2 \qquad \text{for all }x\in U\text{ and }\zeta \in \mathbb C^{2n\times N}. \]
(A term of the form $\int e^ i_{IJ}v_ I\overline{\partial _ iw_ J}$ can be integrated by parts into terms of the last two kinds, so it is also allowed.) Let $k\geq 0$, $F\in H^ k(U)$ and $v\in H^1(U)$, and suppose that $\mathcal A(v,w)=\int _ U\sum _ JF_ J\overline{w_ J}\, \mathrm dx$ for all $w\in C^\infty _ c(U)$. Then, for every $\chi \in C^\infty _ c(U)$, the function $\chi v$, extended by zero, belongs to $H^{k+2}(\mathbb R^{2n})$, and $\| \chi v\| _{H^{k+2}(\mathbb R^{2n})}\leq C(\| F\| _{H^ k(U)}+\| v\| _{H^1(U)})$, where $C$ depends only on $\chi $, $k$, $\lambda $ and bounds for the coefficients and their derivatives.
Proof.
In this proof $H^ j$ stands for $H^ j(\mathbb R^{2n})$. We prove the case $k=0$ with difference quotients and then argue by induction on $k$. Let $0\leq \chi \leq 1$, let $h_0$ be half the distance from $\operatorname{Supp}\chi $ to $\partial U$, and for a coordinate direction $e_ l$ and $0<|h|<h_0$ put $\delta _ hv(x)=(v(x+he_ l)-v(x))/h$. For smooth $f$ we have $\delta _ hf(x)=\int _0^1\partial _ lf(x+the_ l)\, \mathrm dt$, so the Cauchy–Schwarz inequality in $t$ and Fubini's theorem bound the $L^2$ norm of $\delta _ hf$ on a set by that of $\partial _ lf$ on a slightly larger set; by mollification the same holds for $f\in H^1$. The function $w=-\delta _{-h}(\chi ^2\delta _ hv)$ lies in $H^1$ and has compact support in $U$. Its mollifications are test functions converging to $w$ in $H^1$ (lemma 2.5.1), and both sides of the equation are continuous in $w$ for the norm of $H^1$, so the equation also holds for $w$.
Discrete integration by parts, $\int f\, \overline{\delta _{-h}g}=-\int (\delta _ hf)\, \overline g$, turns the principal term into $\int \sum \delta _ h(a^{ij}_{IJ}\partial _ jv_ I)\, \overline{\partial _ i(\chi ^2\delta _ hv_ J)}$. Here $\delta _ h(a\, \partial _ jv)=a(\cdot +he_ l)\, \partial _ j\delta _ hv+(\delta _ ha)\, \partial _ jv$ and $\partial _ i(\chi ^2\delta _ hv)=\chi ^2\partial _ i\delta _ hv+2\chi \, \partial _ i\chi \, \delta _ hv$. The ellipticity condition at $x+he_ l$ bounds the main term $\int \sum a^{ij}_{IJ}(x+he_ l)\chi ^2\partial _ j\delta _ hv_ I\overline{\partial _ i\delta _ hv_ J}$ from below by $\lambda \| \chi \nabla \delta _ hv\| _{L^2}^2$. The remaining terms, the lower-order terms and the right-hand side are estimated with $|\delta _ ha|\leq C$, with $\| \delta _ hv\| _{L^2(\operatorname{Supp}\chi )}\leq \| \partial _ lv\| _{L^2(U)}$, and with $\| w\| _{L^2}\leq \| \partial _ l(\chi ^2\delta _ hv)\| _{L^2}\leq \| \chi \nabla \delta _ hv\| _{L^2}+C\| v\| _{H^1(U)}$, where the difference quotients are bounded as above. This gives
\[ \lambda \| \chi \nabla \delta _ hv\| _{L^2}^2 \leq C\bigl (\| F\| _{L^2(U)}+\| v\| _{H^1(U)}\bigr ) \bigl (\| \chi \nabla \delta _ hv\| _{L^2}+\| v\| _{H^1(U)}\bigr ), \]
and hence $\| \chi \nabla \delta _ hv\| _{L^2}\leq C(\| F\| _{L^2(U)}+\| v\| _{H^1(U)})$, independently of $h$.
If $\phi \in C^\infty _ c$ is supported in the interior of $\{ \chi =1\} $, then $-\int \partial _ jv\, \partial _ l\phi =\lim _{h\to 0}\int (\delta _ h\partial _ jv)\, \phi $ is bounded by this constant times $\| \phi \| _{L^2}$. By the Riesz representation theorem in $L^2$ (proposition 6.2.1), applied to the extension of $\phi \mapsto -\int \partial _ jv\, \partial _ l\phi $ to $L^2$, the distributional derivatives $\partial _ l\partial _ jv$ are square integrable there, with the same bound. Since $\chi $ may be chosen equal to one near the support of any given cutoff, this proves the lemma for $k=0$.
Suppose the lemma holds for some $k\geq 0$, let $F\in H^{k+1}(U)$, and fix $\chi $ and an open set $U'\Subset U$ containing $\operatorname{Supp}\chi $. By the lemma for $k$, $v\in H^{k+2}(U')$, with norm at most $C(\| F\| _{H^ k(U)}+\| v\| _{H^1(U)})$. Fix $m$ and test the equation with $-\partial _ mw$, $w\in C^\infty _ c(U')$. Integrating by parts in $x_ m$, we find that $u=\partial _ mv\in H^1(U')$ satisfies $\mathcal A(u,w)=\int \sum _ JF'_ J\overline{w_ J}\, \mathrm dx$ for all $w\in C^\infty _ c(U')$, where, with the indices $I,J$ suppressed,
\[ F'=\partial _ mF+\sum _{i,j}\partial _ i\bigl ((\partial _ ma^{ij})\partial _ jv\bigr )-\sum _ j(\partial _ mb^ j)\partial _ jv-(\partial _ mc)v. \]
Since $v\in H^{k+2}(U')$, we have $F'\in H^ k(U')$ with $\| F'\| _{H^ k(U')}\leq C(\| F\| _{H^{k+1}(U)}+\| v\| _{H^{k+2}(U')})$. The lemma for $k$ on $U'$ gives $\chi \, \partial _ mv\in H^{k+2}$ for every $m$, hence $\chi v\in H^{k+3}$ with the asserted bound.
$\square$
Pome
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