Existence and regularity of monotone solutions to a free boundary problem

• Main Author: De Silva, Daniela
• Other Authors: David Jerison.
• Format: Others
• Language: English
• Published: Massachusetts Institute of Technology 2006
• Subjects: Mathematics.
• Online Access: http://hdl.handle.net/1721.1/31160

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005. === Includes bibliographical references (p. 71-72). === In the first part of this dissertation, we provide the first example of a singular energy minimizing free boundary. This singular solution occurs in dimension 7 and higher, and in fact it is conjectured that there are no singular minimizers in dimension lower than 7. Our example is the analogue of the 8-dimensional Simons cone in the theory of minimal surfaces. The minimality of the Simons cone is closely related to the existence of a complete minimal graph in dimension 9, which is not a hyperplane. The first step toward solving the analogous problem in the free boundary context, consists in developing a local existence and regularity theory for monotone solutions to a free boundary problem. This is the objective of the second part of our thesis. We also provide a partial result in the global context.. === by Daniela De Silva. === Ph.D.