= Solution
{c}
For a sufficiently regular solution, put $w=-\Delta u$, with $\Delta$ the <Laplacian>. Then $-\Delta u-w=0$ and $u-\Delta w=u+\Delta^2u=f$. The zero <Dirichlet boundary conditions> on $u$ and $\Delta u$ become zero <Dirichlet boundary conditions> on $u$ and $w$. Conversely, eliminating $w$ from these two equations recovers the fourth-order equation with the <biharmonic operator>. For the <weak solution> we will construct, these interior identities hold as <distributional identities>; zero <Sobolev traces> are imposed on $u$ and $w$, without assuming classical fourth <derivatives> of $u$.
Let $V=H_0^1(\Omega)\times H_0^1(\Omega)$, the <Cartesian product> of two <zero-boundary Sobolev spaces>, with <norm>
$$
\|(p,q)\|_V^2=\|p\|_1^2+\|q\|_1^2.
$$
This is a <Hilbert space>. Applying <integration by parts> separately to the two equations gives the <weak formulation>
$$
\boxed{\begin{aligned}
(\nabla u,\nabla v)_0-(w,v)_0&=0&&\text{for every }v\in H_0^1(\Omega),\\
(u,z)_0+(\nabla w,\nabla z)_0&=(f,z)_0&&\text{for every }z\in H_0^1(\Omega).
\end{aligned}}
$$
Equivalently, for $U=(u,w)$ and $T=(v,z)$, define the <bilinear form> and <linear functional>
$$
\begin{aligned}
b((p,q),(v,z))&=(\nabla p,\nabla v)_0+(\nabla q,\nabla z)_0-(q,v)_0+(p,z)_0,\\
F(v,z)&=(f,z)_0.
\end{aligned}
$$
We seek $U\in V$ with $b(U,T)=F(T)$ for every $T\in V$.
The <skew-coupled elliptic system> has a useful <bounded bilinear form> estimate with constant one. Indeed, write the <integrand> as the <dot product> of $(\nabla p,\nabla q,-q,p)$ and $(\nabla v,\nabla z,v,z)$. The <Cauchy-Schwarz inequality> gives
$$
|b((p,q),(v,z))|\leq\|(p,q)\|_V\|(v,z)\|_V,
\qquad |F(v,z)|\leq\|f\|_0\|(v,z)\|_V.
$$
For equal arguments, the two zeroth-order coupling terms cancel. If $K$ is the constant in the supplied <Poincaré inequality>, then
$$
b((p,q),(p,q))=\|\nabla p\|_0^2+\|\nabla q\|_0^2
\geq\frac1{1+K}\|(p,q)\|_V^2.
$$
Thus $b$ is a <coercive bilinear form>, although it is not a <symmetric bilinear form>. The <Lax-Milgram theorem> applies and yields
$$
\boxed{\text{There is exactly one }(u,w)\in H_0^1(\Omega)\times H_0^1(\Omega)\text{ solving the weak system.}}
$$
In particular, $\|(u,w)\|_V\leq(1+K)\|f\|_0$. Testing with compactly supported smooth <test functions> identifies $-\Delta u=w$ and $-\Delta w=f-u$ as <distributional identities>. This gives the asserted relation to the original <boundary value problem> at the available regularity.
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