Interior $H^{k+2}$ elliptic regularity

Assume Countable Choice. Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open, $n\ge 1$, $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$, let $k\ge 0$, and let $L,a$ be as in Uniformly elliptic divergence-form operators and their sesquilinear forms with $a^{ij}\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{k+1,\mathrm{\infty }}(\mathrm{\Omega })$, $b^i,c\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{k,\mathrm{\infty }}(\mathrm{\Omega })$, all derivatives bounded by constants $M_{\mathrm{\ell }}$; let $f\in H_{\mathrm{l}\mathrm{o}\mathrm{c}}^k(\mathrm{\Omega })$ and let $u\in H^1(\mathrm{\Omega })$ be a local weak solution of $Lu=f$ (Local weak solutions of a divergence-form operator). Then $u\in H_{\mathrm{l}\mathrm{o}\mathrm{c}}^{k+2}(\mathrm{\Omega })$, and for all open sets ${\mathrm{\Omega }}^{\mathrm{\prime }}⋐{\mathrm{\Omega }}^{\mathrm{\prime }\mathrm{\prime }}⋐\mathrm{\Omega }$ there is $C=C(n,\theta ,k,M_0,\dots ,M_{k+1},{\mathrm{\Omega }}^{\mathrm{\prime }},{\mathrm{\Omega }}^{\mathrm{\prime }\mathrm{\prime }})$ with $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{k+2}({\mathrm{\Omega }}^{\mathrm{\prime }})}\le C(\mathrm{\parallel }f{\mathrm{\parallel }}_{H^k({\mathrm{\Omega }}^{\mathrm{\prime }\mathrm{\prime }})}+\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2({\mathrm{\Omega }}^{\mathrm{\prime }\mathrm{\prime }})})\mathrm{.}$ The gain is exactly two derivatives; the coefficient regularity required is one order above the data order. For $k=0$ the theorem reduces to Interior $H^2$ regularity for divergence-form equations. The scaffold wrote $\mathrm{\parallel }f{\mathrm{\parallel }}_{H^k(\mathrm{\Omega })}+\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}$ on the right-hand side, which is ill-posed for locally Sobolev data on an unbounded $\mathrm{\Omega }$; the nested formulation is the well-posed local statement.

Hunter's Theorem 4.28 (printed p. 114) states the result with the bound $\mathrm{\parallel }f{\mathrm{\parallel }}_{H^k(\mathrm{\Omega })}+\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}$ for data in $H^k(\mathrm{\Omega })$; the library formulation localises to $H_{\mathrm{l}\mathrm{o}\mathrm{c}}^k$ data on a nested pair, which is the form actually proved by the chain induction. Teschl's Corollary 10.17 and Laugesen's Theorem 5.8 give the same theorem by the same iteration of the interior estimate.