New universal estimates for free boundary problems arising in plasma physics

TL;DR

This paper establishes sharp universal energy estimates and bounds for solutions of free boundary problems in plasma physics, providing new explicit criteria for free boundary existence and solution norms.

Contribution

It introduces the first explicit estimates for superlinear free boundary problems in plasma physics, including positivity thresholds and solution bounds.

Findings

Derived a sharp positivity threshold for free boundary existence.

Established explicit bounds for the $L^{}$-norm of solutions.

Provided necessary conditions for free boundary presence inside the domain.

Abstract

For $\Omega\subset \mathbb{R}^2$ a smooth and bounded domain, we derive a sharp universal energy estimate for non-negative solutions of free boundary problems on $\Omega$ arising in plasma physics. As a consequence, we are able to deduce new universal estimates for this class of problems. We first come up with a sharp positivity threshold which guarantees that there is no free boundary inside $\Omega$ or either, equivalently, with a sharp necessary condition for the existence of a free boundary in the interior of $\Omega$. Then we derive an explicit bound for the $L^{\infty}$-norm of non-negative solutions and also obtain explicit estimates for the thresholds relative to other neat density boundary values. At least to our knowledge, these are the first explicit estimates of this sort in the superlinear case.