  We establish an existence theory for an elliptic boundary value problem in
image analysis known as edge-enhancing diffusion (EED) inpainting. The EED
inpainting problem aims at restoring missing data in an image as steady state
of a nonlinear anisotropic diffusion process where the known data provide
Dirichlet boundary conditions. We prove existence of a weak solution by
applying the Leray-Schauder Fixed point theorem and show that the set of all
possible weak solutions is bounded. Moreover, we demonstrate that under certain
conditions, the sequences resulting from iterative application of the operator
from the existence theory contain convergent subsequences.
