Nonlinear Potential Theory and Harmonic Analysis

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AbstractThe main goal of this project is to develop new methods of nonlinear potential theory, harmonic analysis, and partial differential equations applicable to a wide class of quasilinear and fully nonlinear elliptic equations. Existence of solutions understood in the renormalized or viscosity sense, sharp integral and pointwise estimates of solutions, and removable singularities will be studied. A related part of this project is concerned with nonlinear integral inequalities for Wolff's potentials, as well as inequalities with indefinite weights which appear in the form boundedness problem for linear partial differential operators with distributional coefficients. The proposed research will establish new applications to various problems of nonlinear science, harmonic analysis, operator theory, and mathematical physics which are widely used in the studies of heat transfer, fluid flow, quantum theory, control theory, and stochastic processes. The broader impact will result in further advances in understanding of basic nonlinear phenomena in applied mathematics, physics, environmental and technological sciences which may be useful in teaching and training of graduate students, younger mathematicians and applied scientists.