Nonlinear potential theory, harmonic analysis and integral inequalities

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The main goal of this project is to develop methods of potential theory and harmonic analysis applicable to quasilinear and fully nonlinear partial differential equations and inequalities with singular coefficients and data. This includes existence and regularity of viscosity solutions to elliptic and parabolic problems with nonlinear source terms, sharp estimates of kernels of Neumann series of integral operators with quasimetric kernels, Green's functions and the conditional gauge associated with fractional Schrodinger operators, and extensions of Hessian Sobolev and Poincare inequalities of Trudinger and Wang. Among the tools employed in this project will be dyadic models, discrete Carleson measures, Wolff's potentials, maximal and singular integral operators on non-homogeneous spaces, non-standard duality, and weighted norm inequalities with indefinite weights, along with techniques of modern PDE theory.This research focusing on the interface between several diverse branches of mathematics will advance applications to areas of current interest in differential and integral equations, nonlinear potential theory, conformal geometry, and mathematical physics. In particular, our studies of singularities of solutions will be important to challenging problems where the presence of competing nonlinearities in differential operators and source terms, singularities of the coefficients, data and boundaries of the domains require the development of new analytic techniques. They will result in further understanding of fundamental nonlinear phenomena arising in theoretical physics and applied sciences.