Weak global $W^{2,p}$ regularity for the Dirichlet Laplacian

Initial integrability and the exponent list. Since $u\in H_0^1(\mathrm{\Omega })=W_0^{1,2}(\mathrm{\Omega })$ and $\mathrm{\Omega }$ is a bounded extension domain for $W^{1,2}$ by [F4], the embedding [F3] with $k=1$, $q=2$ gives $u\in L^r(\mathrm{\Omega })$ for every $2\le r\le \frac{2n}{n-2}$ if $n>2$, and for every finite $r$ if $n=2$. Put $q_0:=\min \{p,\frac{2n}{n-2}\}(n>2),q_0:=p(n=2),$ so that $2\le q_0\le p$ and $u\in L^{q_0}(\mathrm{\Omega })$. If $q_0=p$, take the exponent list to be the singleton $Q=\{p\}$ and set $m=0$. Otherwise define a strictly increasing finite list $q_0<q_1<\cdots <q_m=p$ by $q_{i+1}:=\min \{p,\frac{n{q}_i}{n-2q_i}\}$ when $2q_i<n$ and $q_{i+1}:=p$ when $2q_i\ge n$. The list is finite and depends only on $n,p$: while $q_i<p$ and $2q_i<n$ one has $\frac{1}{q_{i+1}}=\frac{1}{q_i}-\frac{2}{n}$ (unless the minimum is $p$, which ends the list), so the reciprocals decrease by the fixed positive amount $\frac{2}{n}$ and the process reaches either $p$ or the region $2q_i\ge n$ after at most $\lceil \frac{n}{2}(\frac{1}{q_0}-\frac{1}{p})\rceil$ steps, after which it reaches $p$ in one more step.

Choice of the shift and the first solve. Let $Q:=\{q_i:0\le i\le m\}$ be the finite set of exponents in the list, so $p\in Q$ and this definition also covers the case $q_0=p$ (then $m=0$ and $Q=\{p\}$). For each $q\in Q$, apply [F2] to $L=\mathrm{\Delta }$ and let ${\lambda }_0(q)$ be its threshold; choose $\lambda >\max (\{0\}\cup \{{\lambda }_0(q):q\in Q\})$. Since $f\in L^p(\mathrm{\Omega })\subseteq L^{q_0}(\mathrm{\Omega })$ and $u\in L^{q_0}(\mathrm{\Omega })$ by step 1.1, the datum $h_0:=f+\lambda u$ lies in $L^{q_0}(\mathrm{\Omega })$; by [F2] there is $z_0\in W^{2,q_0}(\mathrm{\Omega })\cap W_0^{1,q_0}(\mathrm{\Omega })$ with $(\lambda -\mathrm{\Delta })z_0=h_0$ strongly, hence weakly by [F1] (test against compactly supported smooth functions and use density). The weak solution $u$ satisfies the same shifted weak equation with datum $h_0$, as recorded in [F1]. Since $q_0\ge 2$, the space $W_0^{1,q_0}(\mathrm{\Omega })$ is contained in $H_0^1(\mathrm{\Omega })$ (bounded $\mathrm{\Omega }$ gives $W^{1,q_0}(\mathrm{\Omega })\subseteq W^{1,2}(\mathrm{\Omega })$ and the closures transfer), so $z_0-u\in H_0^1(\mathrm{\Omega })$ and [F6] gives $z_0=u$. Hence $u\in W^{2,q_0}(\mathrm{\Omega })\cap W_0^{1,q_0}(\mathrm{\Omega })$.

The bootstrap induction. Suppose $u\in W^{2,q_i}(\mathrm{\Omega })\cap W_0^{1,q_i}(\mathrm{\Omega })$ with $q_i<p$. If $2q_i<n$, then F3 with $k=2$, $q=q_i$ gives $u\in L^r$ for every $r\le \frac{n{q}_i}{n-2q_i}$, in particular $u\in L^{q_{i+1}}$; if $2q_i=n$, then F3 gives $u\in L^r$ for every finite $r$, so $u\in L^p=L^{q_{i+1}}$; if $2q_i>n$, then F3 gives $u\in L^{\mathrm{\infty }}\subseteq L^p=L^{q_{i+1}}$. In all three cases $h_i:=f+\lambda u\in L^{q_{i+1}}$ because $f\in L^p$ and $q_{i+1}\le p$; by [F2] applied at the exponent $q_{i+1}$ there is $z_{i+1}\in W^{2,q_{i+1}}(\mathrm{\Omega })\cap W_0^{1,q_{i+1}}(\mathrm{\Omega })$ solving $(\lambda -\mathrm{\Delta })z_{i+1}=h_i$; by [F1] and [F6], applied exactly as in step 2.1 (with $q_{i+1}\ge q_i\ge 2$ so that $z_{i+1}-u\in H_0^1$), we get $z_{i+1}=u$ and hence $u\in W^{2,q_{i+1}}$. Induction over the finite list gives $u\in W^{2,p}(\mathrm{\Omega })\cap W_0^{1,p}(\mathrm{\Omega })$.

The estimate and almost-everywhere equation. Now that $u\in W^{2,p}(\mathrm{\Omega })\cap W_0^{1,p}(\mathrm{\Omega })$, [F5] applies with $v=u$: $\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{2,p}}\le C_0(\mathrm{\parallel }\mathrm{\Delta }u{\mathrm{\parallel }}_{L^p}+\mathrm{\parallel }u{\mathrm{\parallel }}_{L^p})$. For every $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$, the weak equation [F1] and the definition of the weak Laplacian give ${\int }_{\mathrm{\Omega }}(-\mathrm{\Delta }u)\phi ={\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\cdot \mathrm{\nabla }\phi ={\int }_{\mathrm{\Omega }}f\phi \mathrm{.}$ Both $-\mathrm{\Delta }u$ and $f$ lie in $L^p(\mathrm{\Omega })\subseteq L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega })$, so their regular distributions agree; injectivity [F7] gives $-\mathrm{\Delta }u=f$ almost everywhere. Hence $\mathrm{\parallel }\mathrm{\Delta }u{\mathrm{\parallel }}_{L^p}=\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p}$ and the displayed bound holds with $C=C_0$.

Conclusion. The weak solution $u\in H_0^1(\mathrm{\Omega })$ of $-\mathrm{\Delta }u=f$ with $f\in L^p(\mathrm{\Omega })$, $p>n$, is shown to lie in $W^{2,p}(\mathrm{\Omega })\cap W_0^{1,p}(\mathrm{\Omega })$, and the a priori estimate of [F5] then gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{2,p}}\le C(\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p}+\mathrm{\parallel }u{\mathrm{\parallel }}_{L^p})$ with $C$ depending only on $n,p,\mathrm{\Omega }$. The shifted-equation argument uses the finite sequence of Sobolev exponents and the unique solvability [F2] at each of them; it never assumes $W^{2,p}$ regularity of $u$ in advance.