Optimal sources for elliptic PDEs

We investigate optimal control problems governed by the elliptic partial differential equation $-Δu=f$ subject to Dirichlet boundary conditions on a given domain $Ω$. The control variable in this setting is the right-hand side $f$, and the objective is to minimize a cost functional that depends simultaneously on the control $f$ and on the associated state function $u$. We establish the existence of optimal controls and analyze their qualitative properties by deriving necessary conditions for optimality. In particular, when pointwise constraints of the form $α\le f\leβ$ are imposed a priori on the control, we examine situations where a {\it bang-bang} phenomenon arises, that is where the optimal control $f$ assumes only the extremal values $α$ and $β$. More precisely, the control takes the form $f=\alpha1_E+\beta1_{Ω\setminus E}$, thereby placing the problem within the framework of shape optimization. Under suitable assumptions, we further establish certain regularity properties for the optimal sets $E$. Finally, in the last part of the paper, we present numerical simulations that illustrate our theoretical findings through a selection of representative examples.