Parameter-dependent coercive elliptic variational states and objectives

Let $V$ be a real Hilbert space, let $\mathrm{\Theta }\subseteq {\mathbb{R}}^m$ be open, and write $V^{\ast }$ for the continuous real dual. A parameter-dependent coercive elliptic variational problem is a family of bounded bilinear forms $a_{\theta }:V\times V\to \mathbb{R}$ and bounded linear functionals ${\mathrm{\ell }}_{\theta }\in V^{\ast }$ such that, on each parameter neighborhood under discussion, $\mathrm{\parallel }{a}_{\theta }\mathrm{\parallel }\le M$ and $a_{\theta }(v,v)\ge \alpha \mathrm{\parallel }v{\mathrm{\parallel }}_V^2$ for fixed $M<\mathrm{\infty }$ and $\alpha >0$. Operator-norm $C^1$ data means that the maps $\theta \mapsto A_{\theta }\in \mathcal{L}(V,V^{\ast })$, $A_{\theta }u:=a_{\theta }(u,\cdot )$, and $\theta \mapsto {\mathrm{\ell }}_{\theta }\in V^{\ast }$ are continuously Fréchet differentiable. A state at $\theta$ is a vector $u_{\theta }\in V$ satisfying $a_{\theta }(u_{\theta },v)={\mathrm{\ell }}_{\theta }(v)$ for every $v\in V$. When a state is specified, a $C^1$ objective $J:V\times \mathrm{\Theta }\to \mathbb{R}$ defines the reduced objective $j(\theta )=J(u_{\theta },\theta )$. This definition alone asserts neither state existence nor uniqueness of an optimizer.