A global solution curve for a class of free boundary value problems arising in plasma physics

TL;DR

This paper investigates the existence, multiplicity, and the global solution curve of a free boundary PDE problem from plasma physics, using continuation methods that are also suitable for numerical implementation.

Contribution

It introduces a continuation-based approach to analyze solutions of a free boundary PDE problem, generalizing one-dimensional pendulum equations and applicable to numerical computations.

Findings

Established existence and multiplicity results for the problem.

Developed a continuation method suitable for numerical analysis.

Connected the PDE problem to pendulum-like equations.

Abstract

We study the existence and multiplicity of solutions and the global solution curve of the following free boundary value problem, arising in plasma physics, see R. Temam [18], and H. Berestycki and H. Brezis [3]: find a function $u(x)$ and a constant $b$, satisfying \[ \Delta u+g(x,u)=p(x) \;\; \mbox{in $D$} \] \[ u \,| \, _{\partial D}=b, \;\;\;\; \int_{\partial D} \frac{\partial u}{\partial n} \, ds=0 \,. \] Here $D \subset R^n$, is a bounded domain, with a smooth boundary. This problem can be seen as a PDE generalization of the periodic problem for one-dimensional pendulum-like equations. We use continuation techniques. Our approach is suitable for numerical computations.