I want to solve problems of the following sort: $$ \min_{y \in L^\infty(\mathbb{R}^d)} \lVert y \rVert_{L^1(\mathbb{R}^d)}~\\\mathrm{s.t.} \cdots.$$ The constraints do not matter yet, they are only supposed to make the problem well-defined and non-trivial.
My question is if you do know some literature concerning minimization on $L^\infty$ or comparable Banach spaces (existence theorems first and foremost).
The problem seems very difficult as $L^\infty$ is not reflexive/separable.