Mathematical Sciences: Regularity Theory for Certain Nonlinear Elliptic Equations Involving Derivatives of Rearrangements of Solutions

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Work will focus on the regularity theory for a class of nonlinear elliptic partial differential equations and associated variational problems involving derivatives of the rearrangement of solutions. The equations, known as Grad differential equations, are a widely used model in the nuclear fusion community. In addition, questions concerning convex symmetrization and interpolation and certain free boundary problems will be treated. These questions arise from models of quasi-static equilibrium of confined toroidal plasmas in thermonuclear fusion and from rotational fluid flow. From a purely mathematical standpoint their analysis involves interesting and important problems in the study of rearrangements of a function, convex symmetrization, free boundary problems and regularity theory for nonlinear nonlocal variational problems. The main emphasis will be on variational problems which can be approximated by sequences of multiphase free boundary problems. Recent work has established convergence criteria for such approximation schemes as well as various regularity results for certain solutions. A new approach for establishing analyticity of weak solutions for convex domains is planned. Part of this effort involves the study of generalizations of Schwarz symmetrization to convex domains as well as other methods for decreasing functionals like the Dirichlet norm. //