A Dirichlet problem with free gradient discontinuity

In this article we prove the existence of strong minimizers for Blake & Zisserman’s functional under Dirichlet boundary condition. The result is obtained by showing partial regularity of weak solutions up to the boundary through blow-up techniques and decay properties for bi-harmonic functions in the half-disk. This research is also motivated by the possibility of studying variational formulation of the inpainting problem for 2-dimensional images which are locally damaged and its approximation via the variational convergence introduced and studied by E. De Giorgi.