*Accepted Paper*

**Inserted:** 1 jul 2020

**Last Updated:** 19 may 2021

**Journal:** Proc. Amer. Math. Soc.

**Year:** 2021

**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.

**Keywords:**
free boundary problems, plasma physics, universal estimates

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