Un modèle non local en physique des plasmas: résolution par une méthode de degré topologique

Abstract

Under a few assumptions on the functiong, we prove the existence of the following-free boundary problem: Findu inH ²(Ω) satisfying:

$$\begin{gathered} \Delta u \in g(x, u(x)), [\delta (u)(x), \bar \delta (u)(x)] in \{ u< 0\} \hfill \\ \Delta u = 0 in \{ u \geqslant 0\} \hfill \\ u = constant (but unknown) on the boundary \partial \Omega of \Omega \hfill \\ \int_{\partial \Omega } {\frac{{\partial u}}{{\partial n}}d\sigma = I > 0} (given number) \hfill \\ \end{gathered} $$

where$\delta (u)(x) = meas \{ y \in \Omega , u(x)< u(y)< 0\} ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\delta } (u)(x) = meas\{ y \in \Omega , u(x) \leqslant u(y) \leqslant 0\}$. This is a model of the Grad-Mercier-type describing the equilibrium of a confined plasma in a Tokamak machine.