New Preprint on Dirichlet Boundary Control Problems

12 März 2026

In their new paper “Dirichlet control problems with energy regularization governed by non-coercive elliptic equations”, Thomas Apel, Mariano Mateos, and Arnd Rösch investigate a linear–quadratic Dirichlet control problem governed by a non-coercive elliptic equation posed on possibly non-convex polygonal domains. To stabilize the problem, the authors employ Tikhonov regularization in an energy seminorm and show that the solutions can be described in suitable weighted Sobolev spaces that capture the singularities occurring at the corners of such geometries.

The work builds on the authors’ previous study “Non-coercive Neumann boundary control problems”, which analyzed a linear–quadratic Neumann boundary control problem, including the existence, uniqueness, regularity, and finite element approximation of the state and adjoint equations. In the new contribution, the discretization of the Dirichlet control problem is investigated. To recover optimal convergence rates in non-convex polygonal domains, graded meshes are required; in addition, a discrete projection is introduced to properly handle inhomogeneous boundary conditions. The authors show that the discrete problems remain uniformly strongly convex, which leads to optimal error estimates confirmed by numerical examples.

Thomas Apel, Mariano Mateos, Arnd Rösch: Dirichlet control problems with energy regularization governed by non-coercive elliptic equations. arXiv:2603.09507 [math.OC]