Published Paper

Inserted: 11 apr 2023
Last Updated: 20 mar 2024

Journal: SIAM J. Math. Anal.
Volume: 56
Number: 2
Pages: 2034–2062
Year: 2024
Doi: https://doi.org/10.1137/23M1569873

Abstract:

We consider generalized solutions of the Perona-Malik equation in dimension one, defined as all possible limits of solutions to the semi-discrete 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 subsupersolutions. The monotonicity results are proved for a more general class of evolution curves, that we call $uv$-evolutions.