| Contents |
| By using the notion of relative rearrangement (see, e.g., J.M. Rakotoson;
\emph{R\'{e}arrangement Relatif: Un instrument d'estimations dans les
probl\`{e}mes aux limites}, Math\'{e}matiques et Applications, SMAI, Springer,
Paris 2008.) we introduce a new formulation for some mathematical model
arising in Nuclear Fusion and Image Processing. In particular, we review some
mathematical models related to the stationary regime of a plasma magnetically
confined in nuclear fusion devices such as Tokamaks and Stellarators. We show
that these models, expressed through a nonlinear partial differential
equations (the Grad--Safranov equation), can be reformulated as inverse
problems where several terms of the equations are not a priori known (non
local terms). For the case of Stellarators devices, by using the current
balance within each flux magnetic we reformulate again the problem as a non
local one (see, e.g., J. I. D\'{\i}az, J. F. Padial and J. M. Rakotoson;
\emph{Mathematical treatment of the magnetic confinement in a current carrying
Stellarator},\ Nonlinear Analysis Theory Methods and Applications \textbf{34,}
(1998), 857--887). We will prove some existence and uniqueness result by using
Mini-Max methods. An other new appplication of the relative rearrangement
notion appears in the study of neighbordhood filters in Image Processing. We
shall consider some particular cases of Yaroslavsky filters (L.P. Yaroslavsky
and M. Eden, \emph{Fundamentals of digital optics}, Birkh\"{a}user, Boston,
2003). We take special attention to the ocurrence of flat regions given rised
by the solutions to our problems. |
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