Identification of two-phase free boundary arising in plasma physics

Abstract

We try to estimate the shape and the location of two-phase free boundary which has been studied in [A. Friedman and Y. Liu, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 22 (1995), pp. 375-448] to model a stationary magnetohydrodynamics system. A sufficient condition is obtained to check whether a test disk is included in the plasma region D surrounded by a two-phase free boundary. In the test disk technique, only two simply verifiable conditions are used and indispensableness of the conditions is demonstrated using an example. The technique is applicable to select some of test disks placed in the domain Ω, which gives a rough guess on the shape of plasma region. Next we draw some geometrical properties of plasma region D when the domain Ω possesses a kind of convexity. It is proved that if Ω itself contains the mirror image of the right portion {cursive Greek chi ∈ Ω : cursive Greek chi · ξ > t} of the domain with respect to a line {cursive Greek chi : cursive Greek chi · ξ = t} for all t > t0, then so does the plasma region.