Well-posedness of elliptic control state and adjoint equations

The weak-derivative test identities are linear, and [F4] makes each derivative coordinate intrinsic, so $W^{1,2}(\mathrm{\Omega };\mathbb{R})$ is a vector space with the inner product induced from $L^2(\mathrm{\Omega }{)}^{d+1}$; it induces the displayed full Sobolev norm and is definite because its zeroth coordinate is $v$. Let $(v_n)$ be Cauchy in this norm. Its function coordinates and each of its finitely many weak-derivative coordinates are Cauchy in $L^2(\mathrm{\Omega })$ because each coordinate norm is bounded by the full $W^{1,2}$ norm. By [F3], there are $v,g_1,\dots ,g_d\in L^2(\mathrm{\Omega })$ with $v_n\to v$ and $D_i{v}_n\to g_i$ in $L^2$ for each $i$.

For every $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ and $i$, the weak-derivative identity for $v_n$ is ${\int }_{\mathrm{\Omega }}{v}_n{D}_i\phi =-{\int }_{\mathrm{\Omega }}(D_i{v}_n)\phi$. By [F5], both test functions are in $L^2$; Cauchy–Schwarz lets both integrals pass to the limits in [F3], giving ${\int }_{\mathrm{\Omega }}v{D}_i\phi =-{\int }_{\mathrm{\Omega }}{g}_i\phi$. Thus each $g_i$ is a weak derivative of $v$. By uniqueness in [F4], $D_{i}v=g_i$ as an $L^2$ class, and the component convergences from step 1.1 give $v_n\to v$ in the full $W^{1,2}$ norm. Hence $W^{1,2}(\mathrm{\Omega };\mathbb{R})$ is complete.

By [F4], $V=H_0^1(\mathrm{\Omega };\mathbb{R})$ is the closure of the linear space $C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{R})$ in $W^{1,2}$. It is therefore a closed linear subspace of that complete space, and the inherited inner product is exactly the one in Linear coercive elliptic PDE control problem and reduced objective. Thus $V$ is a real Hilbert space under Countable Choice.

For fixed $q$, the functional $F_q(v)=⟨f+Bq,v⟩$ is bounded and linear, with $\mathrm{\parallel }{F}_q{\mathrm{\parallel }}_{V^{\ast }}\le \mathrm{\parallel }f+Bq{\mathrm{\parallel }}_{V^{\ast }}$. Applying [F2] to $a$ and $F_q$ now gives the unique state and the stated bound.

Define the adjoint form $a^{\ast }(p,v):=a(v,p)$. It is bounded with the same constant $M$, and $a^{\ast }(p,p)=a(p,p)\ge \beta \mathrm{\parallel }p{\mathrm{\parallel }}_V^2$, so it is coercive with the same $\beta$. The functional $G_q(v)=(Ju(q)-u_d,Jv{)}_H$ is bounded by $\mathrm{\parallel }J\mathrm{\parallel }\mathrm{\parallel }Ju(q)-u_d{\mathrm{\parallel }}_H\mathrm{\parallel }v{\mathrm{\parallel }}_V$. Apply [F2] to $a^{\ast }$ and $G_q$; the result is the unique $p(q)$ with $a(v,p(q))=G_q(v)$ and the stated bound. No symmetry of $a$ is used.

Steps 1.1, 2.1, and 3.1 establish the real Hilbert structure required by [F2] using Countable Choice through the real $L^2$ completeness result [F3]. Steps 4.1 and 5.1 give the state, adjoint, uniqueness, and both estimates.