Analysis of a fractal boundary: the graph of the Knopp functionReport as inadecuate

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1 Department of Mathematics Riyadh 2 I2M - Institut de Mathématiques de Marseille

Abstract : A usual classification tool to study a fractal interface is the computation of its fractal dimension. But a recent method developed by Y. Heurteaux and S. Jaffard proposes to compute either weak and strong accessibility exponents or local Lp regularity exponents the so-called p-exponent. These exponents describe locally the behavior of the interface. We apply this method to the graph of the Knopp function. The Knopp function itself has everywhere the same p-exponent. Nevertheless, using the characterization of the maxima and minima done by B. Dubuc and S. Dubuc, we will compute the p-exponent of the characteristic function of domain under the graph of F at each point x,Fx and show that p-exponents, weak and strong accessibility exponents change from point to point. Furthermore we will derive a characterization of the local extrema of the function according to the values of these exponents.

Keywords : H ̈older and Lp regularities Knopp function Fractal interface Extrema Weak and strong accessibility exponents Dyadic expansion Extrema.

Author: Mourad Ben Slimane - Clothilde Melot -



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