Gradient estimates for the Perona-Malik equation
نویسندگان
چکیده
We consider the Cauchy problem for the Perona-Malik equation ut = div ( ∇u 1 + |∇u|2 ) in a bounded open set Ω ⊆ R, with Neumann boundary conditions. If n = 1, we prove some a priori estimates on u and ux. 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. Mathematics Subject Classification 2000 (MSC2000): 35B45, 35B50, 35K55.
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