Proposal: DMS-9801282 Principal Investigator: Albert Baernstein II Abstract: Professor Baernstein is writing a book on the theory and practice of symmetrization, in which particular symmetrization tools such as inequalities for Dirichlet integrals and convolutions will be obtained in a unified way from certain "main inequalities." The tools will then be applied to prove a number of results in real and complex analysis and in partial differential equations. Baernstein is also investigating particular extremal problems in which symmetries of various kinds play a role. Examples include finding the exact values of the Bloch and Landau constants in complex function theory, determining L-p norms of k-plane transforms, and investigating a conjectural integral inequality for gradients of vector fields in the plane whose truth would tell us the L-p norm of a singular integral operator related to quasiconformal mappings, and whose falsity would confirm a conjecture of C.B. Morrey in the calculus of variations which asserts that rank-one convex functions need not be quasisconvex. The prototypical result on which Baernstein's project is modelled is the isoperimetric inequality. In simplest form, this inequality asserts that if a plot of land is to be enclosed by a fence of specified length, then to make the area inside the fence as large as possible, one should lay out the fence in a circle. Each of the unsolved problems mentioned in the first paragraph can be interpreted as an "extremal problem" of the same flavor as the isoperimetric inequality, in that the expected extremals are the competitors which appear to have the most symmetry. The challenge to the researcher is to devise mathematical tools which enable one to prove in a rigorous fashion that the expected extremals really are extremal, or to discover "counterexamples" which show that the established guess for the extremals was wrong. The problems mentioned above come from mathematical fields with connections to various domains of application . Phenomena encountered in the search for exact Bloch and Landau constants are related to crystallography, k-plane transforms occur in tomography, and quasiconformal mappings are related to elasticity theory and to problems about optimal mixtures of materials.
