for $u \in V$, $\mu \in {\mathcal{P}}$ we denote the *partial derivative* of ${\mathcal{J}}(u, \mu)$ w.r.t. the $i$-th component of $\mu$ by $d_{\mu_i}{\mathcal{J}}(u, \mu)$ for $1 \leq i \leq P$. Note that $d_{\mu_i}{\mathcal{J}}(u, \mu) = \partial_\mu {\mathcal{J}}(u, \mu)\cdot e_i $, where $e_i \in {\mathbb{R}}^P$ denotes the $i$-th canonical unit vector. Furthermore, we denote the *gradient* w.r.t. its second argument – the vector of components $d_{\mu_i}{\mathcal{J}}(u, \mu)$ – by the operator $\nabla_\mu {\mathcal{J}}: V \times {\mathcal{P}}\to {\mathbb{R}}^P$. Similarly, if ${\mathcal{J}}$ is Fréchet differentiable w.r.t. each $u \in V$, for each $u \in V$ and each $\mu \in {\mathcal{P}}$ there exists a bounded linear functional $\partial_u {\mathcal{J}}(u, \mu)\in V'$, such that the Fréchet derivative of ${\mathcal{J}}$ w.r.t. its first argument in any direction $v \in V$ is given by $\partial_u {\mathcal{J}}(u, \mu)[v]$. We refer to $\partial_u {\mathcal{J}}(u, \mu)$ simply as the derivative w.r.t. $u$. If ${\mathcal{J}}$ is twice Fréchet differentiable w.r.t. each $\mu \in {\mathcal{P}}$, we denote its *hessian* w.r.t. its second argument by the operator ${\mathcal{H}}_\mu {\mathcal{J}}: V \times {\mathcal{P}}\to {\mathbb{R}}^{P \times P}$.

We treat $a$, $l$, $j$ and $k$ in a similar manner, although, for notational compactness, we indicate their parameter-dependency differently. For instance, interpreting the bilinear form $a$ as 