Boundary Control of Elliptic Solutions to Enforce Local Constraints

We present a constructive method to devise boundary conditions for solutions of second-order elliptic equations so that these solutions satisfy speci c qualitative properties such as: (i) the norm of the gradient of one solution is bounded from below by a positive constant in the vicinity of a nite number of prescribed points; and (ii) the determinant of gradients of n solutions is bounded from below in the vicinity of a nite number of prescribed points. Such constructions nd applications in recent hybrid medical imaging modalities. The methodology is based on starting from a controlled setting in which the constraints are satis ed and continuously modifying the coe cients in the second-order elliptic equation. The boundary condition is evolved by solving an ordinary di erential equation (ODE) de ned so that appropriate optimality conditions are satis ed. Unique continuations and standard regularity results for elliptic equations are used to show that the ODE admits a solution for su ciently long times.