Mathematical Sciences: Non-linear partial differential equations

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This project will study properties of the level sets of solutions to nonlinear partial differential equations, regularity of weak solutions to Monge-Ampere equations, and probabilistic perturbations of free boundary value problems. The principal investigator has preliminary results in all areas to be studied. The techniques to be used borrow, in particular, from techniques developed by the PI in the study of variational inequalities and general regularity theory. The problems to be studied are the next frontier in partial differential equations; they are motivated by problems in areas as diverse as combustion theory and elasticity.