Gradient estimates for the Perona–Malik equation

Abstract

We consider the Cauchy problem for the Perona-Malik equationin a bounded open set ⊆ R n , with Neumann boundary conditions.If n = 1, we prove some a priori estimates on u and u x . Then we consider the semi-discrete scheme obtained by replacing the space derivatives by finite differences. Extending the previous estimates to the discrete setting we prove a compactness result for this scheme and we characterize the possible limits in some cases. Finally, for n > 1 we give examples to show that the corresponding estimates on ∇u are in general false.

A concept of solution and numerical experiments for forward-backward diffusion equations

“…All this is confirmed by the numerical experiments in Section 6.2, see in particular Figs 26,29,35. Numerical simulations indicate that wrinkle formation is a rather complex dynamical phenomenon which depends strongly on the structure of the initial datum u.…”

“…Semidiscretization in time has been considered in [26], where convergence to a solution of (1.2) in the sense of Young measures has been proved. See also [24], [29], [53], [56], [17], [43] and the paper [35], where a partial classification of the properties of the limit function as the spatial mesh size goes to zero is given.…”

Upper bounds on the coarsening rate of discrete, ill‐posed nonlinear diffusion equations

“…In particular, there is a lot of recent effort that concerns developing an existence theory [16,15,30,5,11,12,13]. Another topic of research has been appropriately regularized versions of the PDEs [6,3,2,4,1,22].…”

A class of local classical solutions for the one-dimensional Perona-Malik equation

“…From the analytical point of view, classical solutions are naturally welcome. They also give a new light to a priori estimates on smooth solutions [9,12]. Up to now indeed it was suspected that they were likely to be vacuous in the transcritical case.…”

“…Research has developed along three main lines: finding a suitable notion of weak solution consistent with experiments [4,7,9,13,16,17], proving a priori estimates on classical solutions yielding existence or nonexistence results [8,9,12], extending some one-dimensional results to higher dimension [5,9].…”