arXiv 0907.07722009-07-04math.AP

An example of global classical solution for the Perona-Malik equation

Marina Ghisi, Massimo Gobbino

备注：: 38 pages, 3 figures

英文摘要：

We consider the Cauchy problem for the Perona-Malik equation in an open subset of R^{n}, with Neumann boundary conditions. It is well known that in the one-dimensional case this problem does not admit any global C^{1} 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 C^{2,1} with a transcritical initial condition.

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