Bootstrapping a smooth Poisson problem

Given: The Axiom of Choice and Countable Choice; an open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$; $f\in C^{\mathrm{\infty }}(\mathrm{\Omega })$; and a local weak solution $u\in H^1(\mathrm{\Omega })$ of $-\mathrm{\Delta }u=f$ on $\mathrm{\Omega }$.

A class $u\in H^1(\mathrm{\Omega })$ is a local weak solution of $-\mathrm{\Delta }u=f$ when $a(u,v)={\int }_{\mathrm{\Omega }}f\bar{v}dx$ for every $v\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$, with $a$ the form of the divergence-form operator; by the closure definition this is equivalent to the same identity for every bounded ${\mathrm{\Omega }}_2⋐\mathrm{\Omega }$ and every $v\in H_0^1({\mathrm{\Omega }}_2)$. (Local weak solutions of a divergence-form operator)

Assume Countable Choice. The Laplacian $L=-\mathrm{\Delta }$ has constant coefficients $a^{ij}={\delta }^{ij}$, $b=c=0$, hence $a^{ij}\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{k+1,\mathrm{\infty }}(\mathrm{\Omega })$ and $b,c\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{k,\mathrm{\infty }}(\mathrm{\Omega })$ for every $k\ge 0$ with zeroth-order principal bound $M_0=1$ and all positive-order derivative bounds zero; it is uniformly elliptic with $\theta =1$. (Uniformly elliptic divergence-form operators and their sesquilinear forms)

Assume Countable Choice. Let $k\ge 0$, let $L$ have $a^{ij}\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{k+1,\mathrm{\infty }}(\mathrm{\Omega })$ and $b^i,c\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{k,\mathrm{\infty }}(\mathrm{\Omega })$ with all coefficient derivatives through the indicated orders 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 $Lu=f$. Then $u\in H_{\mathrm{l}\mathrm{o}\mathrm{c}}^{k+2}(\mathrm{\Omega })$. (Interior $H^{k+2}$ elliptic regularity)

Assume the Axiom of Choice. For $n\ge 2$, let ${\mathrm{\Omega }}_0$ be a bounded extension domain in ${\mathbb{R}}^n$, let $k\ge 1$, $1\le p<\mathrm{\infty }$ with $kp>n$. If $m\ge 0$ and $0<\alpha <1$ satisfy $m+\alpha <k-n\mathrm{/}p$, then every $u\in W^{k,p}({\mathrm{\Omega }}_0)$ has a representative in $C^{m,\alpha }(\overline{{\mathrm{\Omega }}_0})$ with norm bounded by $C\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{k,p}({\mathrm{\Omega }}_0)}$. In particular, $p=2$ and an integer $k>n\mathrm{/}2$ give a continuous representative on ${\mathrm{\Omega }}_0$. (Higher-order Sobolev embedding)

Every open ball in ${\mathbb{R}}^n$, $n\ge 2$, is a bounded extension domain: it is a bounded $C^{\mathrm{\infty }}$ domain in the graph sense, so Bounded C^k domains admit integer-order Sobolev extension supplies an extension operator. For $n=1$, on each bounded open interval $I⋐\mathrm{\Omega }$ every class in $W^{1,2}(I)$ has a unique continuous, locally absolutely continuous representative by One-dimensional $W^{1,p}$ functions have unique absolutely continuous representatives. (Sobolev extension domains and extension operators)

Assume Countable Choice. If $u\in H_{\mathrm{l}\mathrm{o}\mathrm{c}}^2(\mathrm{\Omega })$ is a local weak solution of $Lu=f$ with $a^{ij}\in W^{1,\mathrm{\infty }}(\mathrm{\Omega })$, bounded first coefficient derivatives, and $f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2(\mathrm{\Omega })$, then the equation holds pointwise almost everywhere, with $D_i(a^{ij}{D}_{j}u)$ understood through the a.e. defined product $(D_i{a}^{ij})D_{j}u+a^{ij}{D}_i{D}_{j}u$; for the constant coefficients of the Laplacian this is the a.e. identity $-{\sum }_i{D}_i{D}_{i}u=f$. (Interior $H^2$ regularity for divergence-form equations)