{"ID":2875957,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.01521","arxiv_id":"2509.01521","title":"Optimal sources for elliptic PDEs","abstract":"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.","short_abstract":"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 co...","url_abs":"https://arxiv.org/abs/2509.01521","url_pdf":"https://arxiv.org/pdf/2509.01521v1","authors":"[\"Giuseppe Buttazzo\",\"Juan Casado-Díaz\",\"Faustino Maestre\"]","published":"2025-09-01T14:42:40Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}