# Fully nonlinear equations with applications to grad equations in plasma physics ; Interaction between a one phase free boundary problem and an obstacle problem ; Optimal trace Sobolev inequalities

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In this thesis we address three different problems. First, we prove existence and regularity for a fully nonlinear and nonlocal equation which arises in plasma physics. This is a generalization of Grad Equations which model the behavior of plasma confined in a toroidal vessel called TOKAMAK. We prove existence of a W [superscript 2,p]-viscosity solution and regularity up to C [superscript 1, alpha] [Ω overlined] for any α<1. Then we elaborate in how to improve this regularity near the boundary. The main ingredient to study is the nonlocality due to the presence of the measure of the superlevel sets in the equations. Second, we address a problem which models a reaction-diffusion process. Existence is proved, and the solution solves a one phase free boundary problem in the lower half space [doublestruck R] [superscript 3 underscored]. Its trace in [doublestruck R superscript 2] solves an obstacle problem for a given obstacle. We study the exchange between the diffusion in the horizontal plane and the lower half space. Third, we work with a family of variational problems with critical volume and trace constraints. This arises from the study of "best p-Sobolev inequalities" for n≥2 and p is an element of. We extend the analysis from [MV05] and [MN17] for an open set Ω [is a proper subset of] [doublestruck R superscript n]. We prove existence of minimizers for Ω bounded and n>2p, and existence of generalized minimizers for n>p. We also establish rigidity results for the comparison theorem "balls have the worst best Sobolev inequalities" from [MV05].