Boundary control of elliptic solutions to enforce local constraints

TL;DR

This paper introduces a constructive boundary control method for elliptic equations to ensure solutions meet specific local gradient and determinant constraints, with applications in hybrid medical imaging.

Contribution

It develops a novel ODE-based approach to design boundary conditions that enforce qualitative properties of elliptic solutions, advancing control techniques in PDEs.

Findings

Successfully enforces gradient bounds near prescribed points

Ensures lower bounds on determinants of multiple solutions' gradients

Provides a method applicable to hybrid medical imaging

Abstract

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.