Let $A : \mathcal{D}(X) \subseteq X \to X$ be an unbounded linear operator on some Banach space $X$ of real-valued functions. I am looking for a reference on solutions to optimization problems of the following form:
$$ \mathrm{min}_{f \in \mathcal{F} \subseteq \mathcal{D}(A)} (Af)(0), $$ where $\mathcal{F}$ is some convex subset of $\mathcal{D}(A)$.
Example problem: Take $X = L_{2}(\mathbb{R}^{2})$, $Af = \mathrm{div}f = \partial_{x_1}f + \partial_{x_2}f$, $\mathcal{D}(A) = C^{1}(\mathbb{R}^{2})$ and $\mathcal{F}$ is some abstract convex subset of $C^{1}(\mathbb{R}^{2})$.