Young measure solutions and instability of the one-dimensional Perona-Malik equation

We apply a variational approach to the one-dimensional version of the widely used Perona-Malik equation in image processing. We rephrase the problem into the one related to the quasiconvex hull of a graph in the space of 2×2 matrices M2×2. We then use the solutions of some heat equations as the centre of the mass for the Young measure-valued solutions to construct the approximate solutions by using simple laminates. The approximate solutions can be viewed as solutions of a perturbation problem by W−1,p (or W−1,∞) functions. The sequences of the approximate solutions generates Young measure-valued solutions. Our results also show that the solutions of the one-dimensional Perona-Malik equation are unstable under small W−1,∞ perturbations.