Problems in Regularity Theory for Nonlinear Partial Differential Equations

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The proposal is aimed at the study of regularity properties of solutionsof various systems of nonlinear partial differential equations. These include:(i) Quasilinear elliptic systems arising as Euler-Lagrange equationsof general multiple integrals in the Calculus of Variations and hyperbolic systems of conservation laws. Questions arising in the study ofregularity of solutions of these equations will be studied from the pointof view of the theory of Compensated Compactness, which provides a unifyingplatform for approaching these seemingly diverse problems. We expect that new methods in Compensated Compactness recently developedby Stefan Muller and the PI will enable us to give answers to some relativelyold open questions.(ii) The Navier-Stokes equations. Here we plan to study both the interiorand boundary regularity for important special classes of solutions.The motivation for these investigation is the following: since regularityquestions for general solutions are notoriously intractable, it will be usefulto study some non-trivial special cases. Hopefully this will provide insights for the general case.We also plan to study the relationship between Morrey's quasiconvexityand local ellipticity conditions for functions on two by two matrices.This problem has recently been linked to some old and difficult conjecturesin Harmonic Analysis and the theory of quasi-conformal mappings, and we expectthat the interaction of ideas from these different areas will be fruitful.The main aim of this proposal is the study of regularity properties ofsolution of nonlinear partial differential equations. Such equationsappear in many models of the real-world processes. The task ofmathematicians is to solve these equations. Often thisis done by a numerical simulation on a computer. To be able to dosuch simulations in a reliable way, one must know something aboutthe qualitative behavior of the solutions. For example, if we expect thatthe solutions will oscillate wildly over small distances, we might needa different method than in the case when the solutions change onlyslowly. Roughly speaking, one of the main task of regularitytheory is to develop methods which would enable us to reliablypredict the behavior of the solutions, so that we can then chooseadequately the methods for solving the equations. The proposed research will hopefully improve our understandingof these difficult and important questions.