Free boundary problems of variational type

This thesis consists of two chapters. In chapter I, we consider an axially symmetric jet flow arising from high speed fiber coating. This problem occurs when we coat fiber optic. As the fiber moves in a high speed through a pressurized tank containing the coating material, there is formed at the entrance of the container a free surface called the upper meniscus. Based on ideal flows, we set up a time-independent model consisting of solving an elliptic equation in the fluid region for the stream function u, and u satisfies the Bernoulli's free boundary conditions.; We prove that this coating problem has a unique solution. Various properties of the free boundary are also discussed and obtained. The tools I used in the proof are based in part on the variational approach for jet and cavity flows introduced by Alt, Caffarelli and Friedman, and on Serrin's under-over theorem concerning the geometric properties of free boundaries of ideal flows.; In chapter II, we consider a free boundary problem arising from plasma physics. In the Tokamak machine, the thermonuclear plasma is confined inside a perfect superconducting shell. The boundary of the plasma region is unknown in advance. A 2-dimensional model describing the equilibrium plasma subject to a surface current leads to an interior free boundary problem of Bernoulli type in an annular region. On the inner (unknown) boundary of the region, the solution of the Laplace equation satisfies the zero Dirichlet condition and a Neumann-type condition. On the outer (given) boundary, the solution assumes a constant value.; We established the existence and studied uniqueness under some assumptions. Examples of nonuniqueness are also given.; The plasma problem was first posed by Demidov (1978), and remains open since then. Nevertheless, this problem turns out to be closely related to a variational problem with volume constraint studied by Acker, and by Aguilera, Alt and Caffarelli. In the proof of existence, we use this variational problem with an undetermined parameter as an auxiliary problem. In the proof of uniqueness, we use a "folding argument".