  We present a constructive method to devise boundary conditions for solutions
of second-order elliptic equations so that these solutions satisfy specific
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 finite number of
prescribed points; and (ii) the determinant of gradients of $n$ solutions is
bounded from below in the vicinity of a finite number of prescribed points.
Such constructions find applications in recent hybrid medical imaging
modalities.
  The methodology is based on starting from a controlled setting in which the
constraints are satisfied and continuously modifying the coefficients in the
second-order elliptic equation. The boundary condition is evolved by solving an
ordinary differential equation (ODE) defined so that appropriate optimality
conditions are satisfied. Unique continuations and standard regularity results
for elliptic equations are used to show that the ODE admits a solution for
sufficiently long times.
