Nested-domain induction for interior elliptic derivatives

The induction claim is ${\mathrm{P}}_j$: $u\in H^{j+2}({\mathrm{\Omega }}_j)$ with the bound of the Statement, for $0\le j\le k$; the chain and the coefficients are fixed as in the hypotheses, and the sets ${\mathrm{\Omega }}_{-1},\dots ,{\mathrm{\Omega }}_{k+1}$ are nested with all compact inclusions strict.

Base case $j=0$. The chain gives ${\mathrm{\Omega }}_0⋐{\mathrm{\Omega }}_{-1}⋐\mathrm{\Omega }$, so [F4] applies to $u$ on the pair $({\mathrm{\Omega }}_0,{\mathrm{\Omega }}_{-1})$ (the datum $f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2(\mathrm{\Omega })$ restricts to $L^2({\mathrm{\Omega }}_{-1})$, and the coefficient bounds $M_0,M_1$ are the ones in [F2] for $k\ge 0$, the case $k=0$ reading $a^{ij}\in W^{1,\mathrm{\infty }}$ and $b,c\in L^{\mathrm{\infty }}$): $u\in H^2({\mathrm{\Omega }}_0)$ with $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^2({\mathrm{\Omega }}_0)}\le C(\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2({\mathrm{\Omega }}_{-1})}+\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2({\mathrm{\Omega }}_{-1})})$, which is ${\mathrm{P}}_0$.

Induction step. Assume ${\mathrm{P}}_{j-1}$ for some $1\le j\le k$, so $u\in H^{j+1}({\mathrm{\Omega }}_{j-1})$ with $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{j+1}({\mathrm{\Omega }}_{j-1})}\le C_{j-1}(\mathrm{\parallel }f{\mathrm{\parallel }}_{H^{j-1}({\mathrm{\Omega }}_{-1})}+\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2({\mathrm{\Omega }}_{-1})})$. Since ${\mathrm{\Omega }}_{j-1}\subseteq {\mathrm{\Omega }}_{-1}$ and ${\mathrm{\Omega }}_{j-1}$ is open with $\overline{{\mathrm{\Omega }}_{j-1}}⋐\mathrm{\Omega }$, [F5] gives $f\in H_{\mathrm{l}\mathrm{o}\mathrm{c}}^k\Rightarrow f\in H^m({\mathrm{\Omega }}_{j-1})$ for every $m\le k$; in particular $f\in H^j({\mathrm{\Omega }}_{j-1})$ and $u\in H^{j+1}({\mathrm{\Omega }}_{j-1})\subseteq H^j({\mathrm{\Omega }}_{j-1})$.

The differentiated equation for a top derivative. Fix $\alpha$ with $\mathrm{\mid }\alpha \mathrm{\mid }=j$. Since $u\in H^{j+1}({\mathrm{\Omega }}_{j-1})$ and $f\in H^j({\mathrm{\Omega }}_{j-1})$, [F3] with $m=j$ and $U={\mathrm{\Omega }}_{j-1}$ makes $w:=D^{\alpha }u\in H^1({\mathrm{\Omega }}_{j-1})$ a local weak solution on ${\mathrm{\Omega }}_{j-1}$ of an equation with the same principal part $a^{ij}$ and datum $g_{\alpha }\in L^2({\mathrm{\Omega }}_{j-1})$ satisfying $\mathrm{\parallel }{g}_{\alpha }{\mathrm{\parallel }}_{L^2({\mathrm{\Omega }}_{j-1})}\le C_j(\mathrm{\parallel }f{\mathrm{\parallel }}_{H^j({\mathrm{\Omega }}_{-1})}+\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{j+1}({\mathrm{\Omega }}_{j-1})})\le C_j^{\mathrm{\prime }}(\mathrm{\parallel }f{\mathrm{\parallel }}_{H^j({\mathrm{\Omega }}_{-1})}+\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2({\mathrm{\Omega }}_{-1})})$, the last step by the bound assumed in step 3.1; the coefficient bounds entering $C_j,C_j^{\mathrm{\prime }}$ are the principal bounds through order $j+1$ and lower-order bounds through order $j$.

Two further derivatives. Choose an intermediate open set $V_j$ with $\overline{{\mathrm{\Omega }}_j}\subset V_j⋐{\mathrm{\Omega }}_{j-1}$; such a set exists because $\overline{{\mathrm{\Omega }}_j}⋐{\mathrm{\Omega }}_{j-1}$. Apply the interior $H^2$ theorem [F4] on the nested pair ${\mathrm{\Omega }}_j⋐V_j⋐{\mathrm{\Omega }}_{j-1}$ to $w=D^{\alpha }u$. Then $w\in H^2({\mathrm{\Omega }}_j)$ and $\mathrm{\parallel }w{\mathrm{\parallel }}_{H^2({\mathrm{\Omega }}_j)}\le C(\mathrm{\parallel }{g}_{\alpha }{\mathrm{\parallel }}_{L^2(V_j)}+\mathrm{\parallel }w{\mathrm{\parallel }}_{L^2(V_j)})$, with $C$ depending on $n,\theta$, the coefficients of $w$'s equation and the pair $({\mathrm{\Omega }}_j,V_j)$. Since $V_j\subseteq {\mathrm{\Omega }}_{j-1}$, the datum and $w$ norms are bounded by those on ${\mathrm{\Omega }}_{j-1}$; inserting the bound of step 4.1 gives $\mathrm{\parallel }{D}^{\alpha }u{\mathrm{\parallel }}_{H^2({\mathrm{\Omega }}_j)}\le C_j^{\mathrm{\prime }\mathrm{\prime }}(\mathrm{\parallel }f{\mathrm{\parallel }}_{H^j({\mathrm{\Omega }}_{-1})}+\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2({\mathrm{\Omega }}_{-1})})$.

Completing the induction. Step 5.1 applies to every multi-index $\alpha$ with $\mathrm{\mid }\alpha \mathrm{\mid }=j$, and there are finitely many of them; summing the finitely many bounds gives $u\in H^{j+2}({\mathrm{\Omega }}_j)$ with $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{j+2}({\mathrm{\Omega }}_j)}\le C_j(\mathrm{\parallel }f{\mathrm{\parallel }}_{H^j({\mathrm{\Omega }}_{-1})}+\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2({\mathrm{\Omega }}_{-1})})$, which is ${\mathrm{P}}_j$, with $C_j$ depending only on $n,\theta ,k$, the principal bounds $M_0,\dots ,M_{j+1}$, the lower-order bounds $M_0,\dots ,M_j$, and the sets ${\mathrm{\Omega }}_{-1},\dots ,{\mathrm{\Omega }}_j$. Together with the base case this proves ${\mathrm{P}}_j$ for every $0\le j\le k$.

Conclusion. For every $0\le j\le k$ the solution satisfies $u\in H^{j+2}({\mathrm{\Omega }}_j)$ with the displayed estimate; in particular the regularity is local and the domains shrink once per induction step, each step gaining exactly two derivatives by the interior $H^2$ theorem applied to the order-$j$ derivative of $u$.