Generic properties of free boundary problems in plasma physics

TL;DR

This paper analyzes the global bifurcation structure of positive solutions in free boundary problems from plasma physics, revealing a dichotomy in solution branch shapes and providing criteria for interior free boundary existence.

Contribution

It introduces a general bifurcation analysis framework for free boundary problems in plasma physics and establishes conditions for interior free boundary existence.

Findings

Solution branches are either similar to the disk model or form simple curves ending at zero boundary density.

A criterion for the existence of interior free boundaries is derived.

Application to a nonlinear eigenvalue problem demonstrates the theory's relevance.

Abstract

We are concerned with the global bifurcation analysis of positive solutions to free boundary problems arising in plasma physics. We show that in general, in the sense of domain variations, the following alternative holds: either the shape of the branch of solutions resembles the monotone one of the model case of the two-dimensional disk, or it is a continuous simple curve without bifurcation points which ends up at a point where the boundary density vanishes. On the other hand, we deduce a general criterion ensuring the existence of a free boundary in the interior of the domain. Application to a classic nonlinear eigenvalue problem is also discussed.