Nonlinear harmonic analysis and partial differential equations of Lane-Emden and Riccati type

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Abstract (Nguyen, 0901083)This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). This project will investigate several problems stemming from a class of quasi-linear and fully nonlinear elliptic equations of Lane-Emden and Riccati type with singular coefficients and measure data, and with nonlinear source terms that involve both solutions and their derivatives. Various methods from nonlinear potential theory, harmonic analysis, and the theory of partial differential equations are proposed to obtain sharp a priori estimates for solutions and their derivatives, to derive capacitary inequalities, and to establish the boundedness of the related nonlinear singular operators. As a consequence, complete characterizations of solvability will be found in different explicit terms including geometric (capacitary) or potential theoretic terms. Moreover, removable singularities of solutions and a more general class of operators and nonlinearities related to subelliptic geometry and nonstandard Sobolev spaces will also be considered.The proposed research is closely related to central problems in nonlinear partial differential equations that require hard analysis and major tools from different branches of mathematics, such as harmonic analysis, nonlinear potential theory, control theory, and differential geometry. Many of the ideas to be developed in this project are new, and they promise to bring new results to bear on applications to diverse problems in both pure and applied mathematics. These applications involve the development of models to describe various phenomena that are studied in physics and engineering, including heat transfer, mass transfer, and fluid flow. The proposed research is tightly integrated with the teaching of graduate students and the interaction of the principal investigator with a network of researchers worldwide, which promotes the exchange of results and ideas between diverse scientific communities.