Mathematical Sciences: Quasiconformal Analysis and Harmonic Integrals with Applications to Nonlinear Elasticity

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9401104 Iwaniec The general area of mathematical research represented by this project is that of nonlinear partial differential equations. The main themes grew out of problems in quasiconformal and quasiregular mappings. The work expanded in recent years to include methods from harmonic analysis, calculus of variations, Sobolev spaces, differential geometry and topology. A major impetus to the current state of quasiconformal mapping was given by work of Sullivan and Donaldson on quasiconformal mappings of four-manifolds. In the course of this research, new differential equations were discovered which in many ways generalize the familiar Cauchy-Riemann system or the Beltrami equation. Basic questions to be studied include that of finding conditions thatensure weakly quasiregularity implies strong quasiregularity. Work will also be done investigating singularities of these mappings and the connection between dimension and removable sets of singularities. Additional efforts will be made to analyze A- harmonic mappings and singular integrals which carry certain algebraic structures, such as Grassmannn or Clifford algebras with a goal of determining dimension-free norms on the integrals when treated as transformations of function spaces. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates onthe accuracy of these approximations. ***