Mathematical Sciences: Free Boundary Problems, Fully Nonlinear Equations, Nonlinear Parabolic Equations and the Navier-Stokes Equations

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The project will cover problems in nonlinear partial differential equations. On free boundary problems, new methods are being developed to study the regularity of free boundaries. Work will be done in establishing "Harnack"-type inequalities for free boundaries along the lines of the corresponding inequalities for linear equations with bounded measurable coefficients by De Giorgi and Krylov. Plans are to study two- phase Stefan problems and minimal surface problems as well. Work will also continue on fully nonlinear equations. A main constraint in the theory is the concavity of the nonlinear function in its dependence on the second derivatives of the unknown function. Plans are to investigate where the coefficients are not necessarily continuous. Efforts will be made to get optimal regularity and uniqueness theorems. The regularity of solutions of nonlinear systems of parabolic equations will be investigated. In addition, work will be done on Navier Stokes equations in three space dimensions. The earlier work with Kohn and Nirenberg suggests that the known solution should be in weak Lebesgue Space. This will be explored along with the possible construction of singular solutions.