Fully Nonlinear Equations with Applications to Grad Equations in Plasma Physics

TL;DR

This paper extends a plasma physics equation to the fully nonlinear case, proving existence and regularity of solutions, and addressing nonlocal challenges involving superlevel set measures.

Contribution

It introduces a fully nonlinear version of a plasma physics equation and establishes existence and regularity results, improving boundary regularity near the domain boundary.

Findings

Proved existence of $W^{2,p}$-viscosity solutions.

Achieved regularity up to $C^{1,eta}$ near the boundary.

Addressed nonlocality due to superlevel set measures.

Abstract

In this paper we generalize an equation studied by Mossino and Temam, to the fully nonlinear case. This equation arises in plasma physics as an approximation to Grad equations, which were introduced by Harold Grad, to model the behavior of plasma confined in a toroidal vessel. We prove existence of a $W^{2,p}$-viscosity solution and regularity up to $C^{1,\alpha}(\overline{\Omega})$ for any $\alpha<1$(we improve this regularity near the boundary). The difficulty of this problem lays on a right hand side which involves the measure of the superlevel sets, making the problem nonlocal.