Mathematical Sciences: Some Non-Classical Problems in Partial Differential Equations with Trace Type Functional Coefficients

自由边界和逆问题出现在许多重要物理现象的建模中。前者的一个例子是在一个区域内确定水/冰界面的位置，该区域边界上的温度是指定的，因此在某些位置是亚临界的，在其他位置是超临界的。相反的问题出现了，例如，试图通过测量物体表面的温度和热流来确定物体内部的热源。本文研究了一类系数依赖于解的泛函的偏微分方程的自由边界和反问题。这个项目考察了解的存在性和唯一性问题，并试图提供对这类微分方程更好的理解。

8901301 Cannon Free boundary and inverse problems arise in the modelling of many physically important phenomena. One example of the former is to determine the location of the water/ice interface in a region where the temperature on the boundary of the region is specified so as to be subcritical in some locations and supercritical in others. Inverse problems arise, for instance, in the attempt to locate the heat source inside an object by measuring the temperature and heat flux at the surface. Free boundary and inverse problems of this type are included in the study of a certain class of partial differential equations whose coefficients may depend on functionals of the solution. This project examines questions of existence and unicity of solutions and seeks to provide a better understanding of this class of differential equations.