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We present a constructive method to devise boundary conditions for solutionsof second-order elliptic equations so that these solutions satisfy specificqualitative properties such as: (i) the norm of the gradient of one solution isbounded from below by a positive constant in the vicinity of a finite number ofprescribed points; and (ii) the determinant of gradients of $n$ solutions isbounded from below in the vicinity of a finite number of prescribed points.Such constructions find applications in recent hybrid medical imagingmodalities.  The methodology is based on starting from a controlled setting in which theconstraints are satisfied and continuously modifying the coefficients in thesecond-order elliptic equation. The boundary condition is evolved by solving anordinary differential equation (ODE) defined so that appropriate optimalityconditions are satisfied. Unique continuations and standard regularity resultsfor elliptic equations are used to show that the ODE admits a solution forsufficiently long times.

Boundary control of elliptic solutions to enforce local constraints