Title: On the Grad-Mercier equation in plasma physics: uniqueness, regularity, and free boundary analysis

 Abstract: In this talk, we will explore the regularity, qualitative properties, and uniqueness of solutions to the boundary value problem ( −∆u(x) = g(|u ≥ u(x)|), in Ω, u = 0, on ∂Ω, (1) where g is a continuous function, Ω is a bounded domain, and |u ≥ u(x)| denotes he Lebesgue measure of the superlevel set |{y ∈ Ω | u(y) ≥ u(x)}| . Equation (1) arises in plasma physics as an approximation to the Grad equations, which were introduced by Harold Grad, to model the behavior of plasma confined in a toroidal vessel. Solutions to (1) naturally develop a plateau at the maximum of u also known as a dead core. The talk will focus on the analysis of the regularity of solutions, particularly near the dead core, and the broader qualitative behavior of the solutions. This work is based on joint research with Antonio Farah and Luis