Nonlinear Sobolev Inequalities, Potential Theory and Harmonic Analysis

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This mathematics research project develops methods of potential theory and harmonic analysis that are applicable to the study of geometric integral inequalities, existence and regularity problems, and estimates of solutions for important classes of fully nonlinear and quasilinear equations and inequalities with singular coefficients and singular data. Significant extensions of Hessian Sobolev and Poincare inequalities of Trudinger and Wang, estimates of solutions in terms of Wolff's potentials, as well as sharp estimates of kernels of Neumann series will be obtained for integral operators, Green's functions and the conditional gauge associated with fractional Schrodinger operators. Among the tools employed will be dyadic models, maximal and singular integral operators on non-homogeneous spaces, non-standard duality, and weighted norm inequalities with indefinite weights, along with diverse techniques from the theory of partial differential equations.This mathematics research project will bring a deeper understanding of fundamental nonlinear and fractal phenomena for complex systems governed by nonlinear laws and nonlocal forces arising in applied sciences and engineering, with potential applications to mathematical modeling of non-Newtonian fluids, heat transfer, elasticity, control theory. The techniques to be employed will bridge several areas of mathematics, such as differential geometry, mathematical physics and conformal geometry. The resulting synergies will enrich the arsenal of analytical tools and available techniques, and will attract more students and young researchers to this dynamic area of mathematics and its applications.