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arXiv · 2504.12548

On the Grad-Mercier equation and Semilinear Free Boundary Problems

Luis Caffarelli·Antonio Farah·Daniel Restrepo
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Abstract

In this paper, we establish regularity and uniqueness results for Grad-Mercier type equations that arise in the context of plasma physics. We show that solutions of this problem naturally develop a dead core, which corresponds to the set where the solutions become identically equal to their maximum. We prove uniqueness, sharp regularity, and non-degeneracy bounds for solutions under suitable assumptions on the reaction term. Of independent interest, our methods allow us to prove that the free boundaries of a broad class of semilinear equations have locally finite $H^{n-1}$ measure.

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Luis Caffarelli, Antonio Farah, Daniel Restrepo. 2025-04-17. On the Grad-Mercier equation and Semilinear Free Boundary Problems. https://arxiv.org/abs/2504.12548

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