An example of global classical solution for the Perona-Malik equation
Abstract
We consider the Cauchy problem for the Perona-Malik equation in an open subset of Rn, with Neumann boundary conditions. It is well known that in the one-dimensional case this problem does not admit any global C1 solution if the initial condition is transcritical, namely when the norm of the gradient of the initial condition is smaller than 1 in some region, and larger than 1 in some other region In this paper we show that this result cannot be extended to higher dimension. We show indeed that for n >= 2 the problem admits radial solutions of class C2,1 with a transcritical initial condition.
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