Monotonicity Properties of Limits of Solutions to the Semidiscrete Scheme for a Class of Perona–Malik Type Equations

We consider generalized solutions of the Perona–Malik equation in dimension one, defined as all possible limits of solutions to the semidiscrete approximation in which derivatives with respect to the space variable are replaced by difference quotients. Our first result is a pathological example in which the initial data converge strictly as bounded variation functions, but strict convergence is not preserved for all positive times and, in particular, many basic quantities, such as the supremum or the total variation, do not pass to the limit. Nevertheless, in our second result we show that all our generalized solutions satisfy some of the properties of classical smooth solutions, namely, the maximum principle and the monotonicity of the total variation. The verification of the counterexample relies on a comparison result with suitable sub-/super-solutions. The monotonicity results are proved for a more general class of evolution curves, that we call uv-evolutions.