The principal investigator proposes to analyze some properties of certain free boundary problems. This proposal is to study the homogenization of these problems. The aim of homogenization is to describe the influence of small variations in the local properties of the medium on the global behavior of the solutions. There has been extensive development in the theory of homogenization over the last thirty years but very little of the work has treated free boundary problems.
Free boundary problems arise naturally in the modeling of biological and physical phenomena involving an interface. The work proposed here will use homogenization to study the shapes of interfaces. Part of the proposal treats the problem of the motion of a drop of liquid on an inclined plane - such as water on a window. This is known to be a difficult mathematical problem and has applications in many areas including the spreading of paint and the retention of pesticide sprays on plant leaves. The PI has studied problems of this type in the analysis of flame propagation and his work has led to an understanding of various issues observed there including the propagation of flame fronts.
