A uniformly coercive elliptic variational state is C1 in its parameter

For each $\theta \in {\mathrm{\Theta }}_0$, apply [F1] to $a_{\theta }$ and ${\mathrm{\ell }}_{\theta }$; this gives a unique $u_{\theta }$ and the bound $\mathrm{\parallel }{u}_{\theta }{\mathrm{\parallel }}_V\le {\alpha }^{-1}\mathrm{\parallel }{\mathrm{\ell }}_{\theta }{\mathrm{\parallel }}_{V^{\ast }}$. Since ${\mathrm{\ell }}_{\theta }$ is continuous, these state norms are locally bounded in $\theta$.

For sufficiently small $h$ with $\theta +h\in {\mathrm{\Theta }}_0$, put $\delta u=u_{\theta +h}-u_{\theta }$, $\delta A=A_{\theta +h}-A_{\theta }$, and $\delta \mathrm{\ell }={\mathrm{\ell }}_{\theta +h}-{\mathrm{\ell }}_{\theta }$. Subtract the two variational equations to get $a_{\theta }(\delta u,v)=\delta \mathrm{\ell }(v)-(\delta A{u}_{\theta +h})(v)$. Taking $v=\delta u$, using coercivity and the dual norm, and dividing by $\mathrm{\parallel }\delta u{\mathrm{\parallel }}_V$ when it is nonzero gives $\mathrm{\parallel }\delta u{\mathrm{\parallel }}_V\le {\alpha }^{-1}(\mathrm{\parallel }\delta \mathrm{\ell }{\mathrm{\parallel }}_{V^{\ast }}+\mathrm{\parallel }\delta A\mathrm{\parallel }\mathrm{\parallel }{u}_{\theta +h}{\mathrm{\parallel }}_V)=O(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$. The same bound is immediate when $\delta u=0$. Thus the state map is locally Lipschitz.

For each direction $h\in {\mathbb{R}}^m$, define $w_h$ by $a_{\theta }(w_h,v)=D{\mathrm{\ell }}_{\theta }[h](v)-(D{A}_{\theta }[h]u_{\theta })(v)$. The right side is a bounded linear functional, so [F1] gives a unique $w_h$. Uniqueness and linearity of the right side in $h$ show $h\mapsto w_h$ is linear.

Write $\delta \mathrm{\ell }=D{\mathrm{\ell }}_{\theta }[h]+r_{\mathrm{\ell }}$ and $\delta A=D{A}_{\theta }[h]+r_A$, where $\mathrm{\parallel }{r}_{\mathrm{\ell }}\mathrm{\parallel }+\mathrm{\parallel }{r}_A\mathrm{\parallel }=o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ by [F2]. Subtract the equation for $w_h$ in step 2.2 from the state-difference equation in step 2.1. With $e_h=\delta u-w_h$, this gives $a_{\theta }(e_h,v)=r_{\mathrm{\ell }}(v)-(r_A{u}_{\theta })(v)-(D{A}_{\theta }[h]\delta u)(v)-(r_A\delta u)(v)$. The first two terms have dual norm $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$; the last two have that order because $\mathrm{\parallel }D{A}_{\theta }[h]\mathrm{\parallel }=O(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$, $\mathrm{\parallel }{r}_A\mathrm{\parallel }=o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$, and $\mathrm{\parallel }\delta u{\mathrm{\parallel }}_V=O(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ by step 2.1. Coercivity yields $\mathrm{\parallel }{e}_h{\mathrm{\parallel }}_V=o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$, proving Fréchet differentiability and the stated derivative equation.

The solution operator $A_{\theta }^{-1}:V^{\ast }\to V$ has norm at most ${\alpha }^{-1}$. For nearby $\theta ,\tilde{\theta }$, the identity $A_{\tilde{\theta }}^{-1}-A_{\theta }^{-1}=A_{\tilde{\theta }}^{-1}(A_{\theta }-A_{\tilde{\theta }})A_{\theta }^{-1}$ shows its operator-norm continuity. The formula in step 2.2, together with continuity of $u_{\theta }$, $D{A}_{\theta }$, and $D{\mathrm{\ell }}_{\theta }$, then shows $D_{\theta }{u}_{\theta }$ is continuous. Hence the state map is $C^1$.