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Higher Critical Points in an Elliptic Free Boundary Problem

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Abstract

We study higher critical points of the variational functional associated with a free boundary problem related to plasma confinement. Existence and regularity of minimizers in elliptic free boundary problems have already been studied extensively. But because the functionals are not smooth, standard variational methods cannot be used directly to prove the existence of higher critical points. Here we find a nontrivial critical point of mountain pass type and prove many of the same estimates known for minimizers, including Lipschitz continuity and nondegeneracy. We then show that the free boundary is smooth in dimension 2 and prove partial regularity in higher dimensions.

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Notes

  1. Here we are using the well known fact that \(\displaystyle \int _{\{u=1\}} |\nabla u|^2 \, \mathrm{d}x = 0\).

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Acknowledgements

David Jerison was supported by NSF Grants DMS 1069225, DMS 1500771, and the Stefan Bergman Trust. This work was initiated while Kanishka Perera was visiting the Department of Mathematics at the Massachusetts Institute of Technology, and he is grateful for the kind hospitality of the department.

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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139, USA

    David Jerison

  2. Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL, 32901, USA

    Kanishka Perera

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  1. David Jerison
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  2. Kanishka Perera
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Correspondence to David Jerison.

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Jerison, D., Perera, K. Higher Critical Points in an Elliptic Free Boundary Problem. J Geom Anal 28, 1258–1294 (2018). https://doi.org/10.1007/s12220-017-9862-8

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