Mathematical Sciences: Non-linear partial differential equations

本课题将研究非线性偏微分方程解的水平集性质、蒙日-安培方程弱解的正则性以及自由边值问题的概率摄动。首席研究员在所有要研究的领域都有初步结果。所使用的技术，特别是借鉴PI在研究变分不等式和一般正则性理论时开发的技术。待研究的问题是偏微分方程的下一个前沿；他们的动力来自燃烧理论和弹性等不同领域的问题。

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.