Poisson's equation with $L^2$ data gains two interior derivatives

Given: The Axiom of Choice (for the existence statement only); a bounded open set $\mathrm{\Omega }\subset {\mathbb{R}}^n$; $f\in L^2(\mathrm{\Omega })$; and a weak solution $u\in H_0^1(\mathrm{\Omega })$ of $-\mathrm{\Delta }u=f$ in the sense of Weak Dirichlet solutions for a divergence-form operator.

For $f\in L^2(\mathrm{\Omega })$ with $\mathrm{\Omega }$ bounded, the weak Dirichlet formulation reads $a(u,v)={\int }_{\mathrm{\Omega }}f\bar{v}dx$ for every $v\in H_0^1(\mathrm{\Omega })$; every such $u$ is a local weak solution of $-\mathrm{\Delta }u=f$ on $\mathrm{\Omega }$, because $C_c^{\mathrm{\infty }}(\mathrm{\Omega })\subseteq H_0^1(\mathrm{\Omega })$ and the local definition tests the smaller class. (Weak Dirichlet solutions for a divergence-form operator, Local weak solutions of a divergence-form operator)

The Laplacian $L=-\mathrm{\Delta }$ is the divergence-form operator with $a^{ij}={\delta }^{ij}$, $b^i=0$, $c=0$: the coefficients are constant, hence in $W^{1,\mathrm{\infty }}(\mathrm{\Omega })$ with $\mathrm{\parallel }D{a}^{ij}{\mathrm{\parallel }}_{\mathrm{\infty }}=0$ and $M_1=0$, uniformly elliptic with $\theta =1$, and $M_a=1$, $M_b=M_c=0$. (Uniformly elliptic divergence-form operators and their sesquilinear forms)

Assume Countable Choice. Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open and let $L,a$ be as in Uniformly elliptic divergence-form operators and their sesquilinear forms with $a^{ij}\in W^{1,\mathrm{\infty }}(\mathrm{\Omega })$, $\mathrm{\parallel }D{a}^{ij}{\mathrm{\parallel }}_{\mathrm{\infty }}\le M_1$, and $f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2(\mathrm{\Omega })$. If $u\in H^1(\mathrm{\Omega })$ is a local weak solution of $Lu=f$ on $\mathrm{\Omega }$, then $u\in H_{\mathrm{l}\mathrm{o}\mathrm{c}}^2(\mathrm{\Omega })$ and for every pair of open sets ${\mathrm{\Omega }}^{\mathrm{\prime }}⋐{\mathrm{\Omega }}^{\mathrm{\prime }\mathrm{\prime }}⋐\mathrm{\Omega }$ the theorem gives the interior estimate of Interior $H^2$ regularity for divergence-form equations. When $f\in L^2(\mathrm{\Omega })$, the estimate on ${\mathrm{\Omega }}^{\mathrm{\prime }}$ is bounded by the global-norm estimate displayed in the statement because ${\mathrm{\Omega }}^{\mathrm{\prime }\mathrm{\prime }}\subseteq \mathrm{\Omega }$; its constant may be written $C(n,\theta ,M_a,M_b,M_c,M_1,{\mathrm{\Omega }}^{\mathrm{\prime }},\mathrm{\Omega })$ after fixing such an intermediate ${\mathrm{\Omega }}^{\mathrm{\prime }\mathrm{\prime }}$ from ${\mathrm{\Omega }}^{\mathrm{\prime }}$ and $\mathrm{\Omega }$.

Assume the Axiom of Choice. If $\mathrm{\Omega }$ is nonempty, every $F\in H^{-1}(\mathrm{\Omega })$ has exactly one weak solution of the Dirichlet problem $-\mathrm{\Delta }u=F$ with zero boundary values; for $F(v)={\int }_{\mathrm{\Omega }}f\bar{v}dx$ with $f\in L^2(\mathrm{\Omega })$ this supplies existence and uniqueness in the example. If $\mathrm{\Omega }=\mathrm{\varnothing }$, then $H^{-1}(\mathrm{\Omega })=\{0\}$ and the zero class is the unique weak solution directly. (Existence and uniqueness for the weak Dirichlet Poisson problem)