Certain Free Boundary Problems

Purdue University, West Lafayette IN

Investigators

Abstract

Certain Free Boundary Problems. Abstract of Proposed Research Arshak Petrosyan This project is to study the properties, especially the regularity, of the free boundary in certain elliptic and parabolic problems. A number of different free boundary problems will be investigated. Most of the problems to be studied in this project arise as the solution of variational problems with nonlocal constraints. Some of the proposed problems are governed by subelliptic operators, which lack the uniform ellipticity, but have a sufficiently rich geometric structure (e.g. in Carnot groups) that allows the development of a satisfactory theory of free boundaries. Other problems resemble traditional variational inequalities, but lack an inequality constraint. For those problems, the main approach is based on application of rather deep monotonicity formulas (sometimes more than one) that allow the proof of results similar to those for variational inequalities associated with obstacle problems. Moreover the free boundaries often have remarkable properties, sometimes akin to those of minimal surfaces. Free boundary problems involve equations (or systems of equations) that are satisfied in a region that is unknown apriori. The boundary of the region where the equation holds is called the free boundary and is generally characterized by extra side conditions. The solution of the problem involves identifying both the solution of the equation and the region with the free boundary. Free boundary problems arise in applications ranging from potential theory and geometry to optimal control, robotics, and superconductivity.