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

该项目将涉及非线性偏微分方程的问题。在自由边界问题上，研究自由边界正则性的新方法正在发展。将沿着De Giorgi和Krylov关于有界可测系数线性方程的相应不等式的路线，建立自由边界的“Harnack”型不等式。计划研究两相斯蒂芬问题和最小曲面问题。我们还将继续研究完全非线性方程。该理论的一个主要约束是非线性函数在依赖于未知函数二阶导数时的凹性。计划是调查哪些系数不一定是连续的。努力得到最优正则性和唯一性定理。研究抛物型非线性方程组解的正则性。此外，我们还将研究三维空间中的Navier Stokes方程。Kohn和Nirenberg的早期工作表明，已知的解决方案应该在弱勒贝格空间中。这将与奇异解的可能构造一起探讨。

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.