The problem considered is that of minimizing a linear function \(L(u(t^1,x))\) on the class of linear functionals satisfying (*) \(u_t = Eu\) and certain initial and boundary conditions on a set \(T\times \Omega_0\) where \(T= [t^0,t^1]\), \(\Omega_0\) is a compact subset in Euclidean space and \(E\in\mathcal E\), a set of linear elliptic operators. Under the assumption that \(\mathcal E\) is weakly closed (defined herein) and the set of solutions to (*) is nonempty, it is shown that a minimizing solution exists. The hypotheses on the set \(\mathcal E\) are essentially those usually imposed on linear parabolic differential equations.