New universal estimates for free boundary problems arising in plasma physics
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- by Daniele Bartolucci and Aleks Jevnikar PDF
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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.References
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Additional Information
- Daniele Bartolucci
- Affiliation: Department of Mathematics, University of Rome “Tor Vergata”, Via della ricerca scientifica n.1, 00133 Roma, Italy
- MR Author ID: 699143
- Email: bartoluc@mat.uniroma2.it
- Aleks Jevnikar
- Affiliation: Department of Mathematics, Computer Science and Physics, University of Udine, Via delle Scienze 206, 33100 Udine, Italy
- MR Author ID: 1037775
- ORCID: 0000-0002-6912-4760
- Email: aleks.jevnikar@uniud.it
- Received by editor(s): November 28, 2020
- Received by editor(s) in revised form: February 5, 2021, March 17, 2021, and April 28, 2021
- Published electronically: November 4, 2021
- Additional Notes: The first author’s research was partially supported by: Beyond Borders project 2019 (sponsored by Univ. of Rome “Tor Vergata”) “Variational Approaches to PDE’s”, MIUR Excellence Department Project awarded to the Department of Mathematics, Univ. of Rome Tor Vergata, CUP E83C18000100006
- Communicated by: Ryan Hynd
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 673-686
- MSC (2020): Primary 35J20, 35J61, 35Q99, 35R35, 76X05
- DOI: https://doi.org/10.1090/proc/15678
- MathSciNet review: 4356177