Existence and regularity of monotone solutions to a free boundary problem
Author(s)De Silva, Daniela
Massachusetts Institute of Technology. Dept. of Mathematics.
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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..
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (p. 71-72).
DepartmentMassachusetts Institute of Technology. Dept. of Mathematics.; Massachusetts Institute of Technology. Department of Mathematics
Massachusetts Institute of Technology