On the Grad-Mercier equation and Semilinear Free Boundary Problems

TL;DR

This work analyzes Grad-Mercier (queer) equations $- abla^2u=g(|u\ge u(x)|)$ arising in plasma physics, revealing a dead core and a free boundary structure. The authors fuse a nonvariational uniqueness argument with a semilinear reformulation $v=\max u-u$, then transform to a degenerate one-phase problem via $w=h(v)$ to leverage Alt-Phillips-type estimates, deriving sharp regularity ($C^{1,1}$; in radial cases $C^{2,1}$) and nondegeneracy, and establishing finite $\mathcal{H}^{n-1}$ measure for a broad class of semilinear free boundaries. They prove uniqueness under $g$ nondecreasing, characterize the dead-core size $|D_u|=\alpha$ under $(H_2)$, and show that in radially symmetric domains the dead core is a homothetic region with cubic detachment from the maximum. The paper also provides a general framework (via (A1)-(A3) and (A2)) guaranteeing finite perimeter of $FB(v)$ for semilinear equations, linking perimeter finiteness to optimal growth bounds on the right-hand side and offering a robust method to analyze non-scale-invariant free-boundary problems in plasma models.

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.

Theorems & Definitions (64)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Definition 2.1
• Lemma 2.2
• proof
• Corollary 2.3